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FIRST    STEPS 

Among    Figures. 

A  Drill  Book  in  the  Fundamental  Rules 
of  Arithmetic. 


TEACHERS'  Edition. 


BY 

LEVI    N.    BEEBE, 

CANANDAIGUA,    N.    Y. 
Sixth  BsmoN,  SKLAResD  and  CABBrt7Li.T  Rxyissd. 


SYRACUSE,    N.   Y. : 

C.  W.  Bardeen,  Publisher. 
1881. 


B4- 


Ck)pyright,  1877,  by  L»ti  N.  Binn. 
EDUCATION  DEFT- 


Oca 


PREFACE  TO  TEACHERS* 
EDITION. 


In  putting  this  work  before  the  public  the 
author  disclaims  any  ambitious  schemes  or 
*•  great  expectations,"  but  he  wishes  to  have 
the  book  for  the  use  of  his  assistant  teachers 
both  as  to  methods  and  examples.  The  author 
has  used  some  parts  of  it  for  many  years  and 
feels  confident  that  excellent  results  may  be 
obtained  by  using  it. 

The  aim  of  the  book  is  to  give  so  much 
practice  as  to  fix  each  method  in  the  pupil's 
mind,  rather  than  to  deal  with  the  philosophy 
of  each  operation,  leaving  any  teacher  who 
believes  that  no  step  should  be  taken  unless 
the  pupil  untlcrstands  the  reasoning  process 
by  which  that  step  may  be  reached,  to  give  it 
in  his  own  way.  It  is  possible  that  a  few  who 
see  this  book  may  have  found  that  7  times  8 
are  56  by  actual  addition,  yet  those  who  have 
never  added  it  may  know  the  fact  just  as  well 
for  all  practical  purposes. 

If  no  one  were  to  eat  until  he  understood 
how  food  nourishes  the  system  there  would  be 
a  deal  of  hunger  in  the  world. 


f,4!.<45 


4  PREFACE  TO  TEACHERS    EDITION. 

This  book  deals  only  with  the  fundamental 
rules  of  arithmetic.  The  intention  is  that  they 
shall  be  so  thoroughly  mastered  that  much  less 
time  will  be  required  for  the  remainder  of  the 
subject  of  arithmetic  than  would  otherwise 
be  needed. 

The  teacher  is  to  use  the  Teachers'  Edition 
for  one  to  two  years  before  the  pupil  has 
advanced  enough  to  use  the  Pupils'  Edition  or 
in  fact  any  book  on  arithmetic.  It  is  recom- 
mended that  teachers  begin  to  teach  numbers 
as  given  in  the  first  part  of  this  book  after 
pupils  who  have  the  alphabet  and  words  to 
learn  have  been  in  school  four  to  six  months. 

In  each  new  operation  the  examples  are 
very  easy  ;  as  more  problems  are  given  they 
gradually  increase  in  difficulty.  By  teaching 
the  four  operations  of  addition,  subtraction, 
multiplication,  and  division,  from  the  first, 
the  examples  are  of  such  a  kind  as  to  compel 
some  thoughtfulness  on  the  part  of  the  pupil. 

Much  pains  has  been  taken  to  make  exam- 
ples of  a  sort  to  interest  the  youngest  pupils. 

Those  teachers  who  wish  to  teach  only 
addition  and  subtraction  at  first  can  designate 
those  examples  involving  multiplication  or 
division  by  some  mark,  and  omitting  them 
may  return  to  them  afterwards  and  so  secure 
the  variety  of  examples  so  essential  to  a  pupil's 
real  progress.  It  has  been  found,  however, 
by  actual  trial  that  pupils  may  learn  the  four 
operations  from  the  first  without  serious 
difficulty. 


PREFACE  TO  TEACHERS    EDITION.  5 

The  first  pages  are  devoted  to  what  is  known 
as  the  "  Grube  Method."  If  the  teacher  pre- 
fers it,  the  schedules  may  be  omitted,  and,  in 
passing  through  the  first  time,  the  multiplica- 
tion and  division  also,  as  before  stated. 

The  author  hopes  that  teachers  into  whose 
hands  this  work  may  come  will  give  it  a 
thorough  examination.  Special  attention  is 
called  to  the  treatment  of  numeration  and 
notation.  The  examples  are  not  all  given  in 
one  place,  to  be  forgotten,  but  are  so  placed  as 
to  review  the  subject  often. 

Attention  is  called  to  the  examples  for  rapid 
solving  and  the  illustration  of  the  easy  exam- 
ples given  under  each  rule.  Also  to  the 
method  of  teaching  long  division  and  to  the 
definition  of  addition. 

'i'he  method  of  teaching  the  addition,  sub- 
traction, multiplication  and  division  tables  is 
believed  to  be  entirely  new,  so  far  as  being 
published  is  concerned.  The  author  dis- 
covered and  used  the  method  about  ten  years 
ago,  and  in  his  school  has  found  it  invaluable. 

To  hear  a  recitation  of  a  large  class  in 
tables  and  make  the  questions  to  each  pupil 
promiscuous,  and  yet  full  enough  to  satisfy  the 
teacher  that  the  pupil  has  a  thorough  knowl- 
edge of  the  tables  gone  over,  is  not  only  very 
wearying  to  the  teacher  but  is  exceedingly- 
difficult  also.  By  the  old  method  a  pupil 
frequently   acquires   the    habit   of  saying   the 


0  PREFACE  TO  TEACHERS'  EDITION. 

table  from  the   beginning  to  find  the  result  of 
any  combination,  as  7  times  6. 

To  enable  any  one  to  make  new  series  like 
those  here  given,  I  insert  the  method.  The 
following  is  for  9's  and  review.  In  the  given 
lines  of  figures  there  is  one  more  figure  in  the 
upper  line  than  in  the  lower  one: 

3456789 

456789 
If  the  upper  line  be  written  several  times 
and  the  lower  line  in  the  same  way  as  follows  : 
34567893456789 


4567 

8 

9 

4 

5     6     7 

8 

9    4     5 

3456 
6789 

7 
4 

8 

5 

9 
6 

3     4     5 
789 

6 
4 

7  8  9 
5    6     7 

3456 
8945 

7 
6 

8 
7 

9 
8 

3     4     5 
9     4     5 

6 
6 

7  8  9 
7    8     9 

The   first 

9 

in 

the 

lower  line 

:  comes  one 

place   before 

the   9   in 

the   upp€ 

IT 

line  ;  the 

second  9  in  the  lower  line  comes  two  places 
before  the  second  9  in  the  upper  line,  and  so 
on  until  it  has  been  under  every  figure  in  the 
upper  line.  If  written  farther,  the  series  wil) 
be  repeated  as  shown  above,  where  4  and  5 
occur  'again  at  the  end  as  they  did  at  the 
beginning.  The  upper  line  of  figures  must  be 
written  one  less  number  of  times  than  there 
are  figures  in  it. 

This  series  may  be  used  for  addition  or 
multiplication,  thus  :  4  and  3  are  7,  5  and  4  are 
9,  &€.,  or  4  times  3  are  12,  5  times  4  are  20,  &c. 


6 

9 

5 

8 

4 

7 

3 

6 

9 

5 

8 

4 

7 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

6 

9 

5 

8 

4 

7 

3 

6 

9 

5 

8 

4 

7 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

6 

9 

5 

8 

4 

7 

3 

6 

9 

5 

8 

4 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

PREFACE  TO  TEACHERS    EDITION.  ^ 

This  arrangement  is  objectionable  for  most 
of  the  results  in  addition  vary  only  by  2  or  4 
and  are  not  as  promiscuous  as  they  should  be. 
By  disarranging  the  upper  line  of  figures  we 
have  6,  9,  5,  8,  4,  7,  3.  Re-writing  this  for  the 
upper  line  and  writing  the  lower  line  as  before 
we  have 

3 

5 
3 

7 

3 
9 

which,  like  the  other  series,  contains  every 
combination  between  4  and  3  and  9  and  9 
inclusive,  and  none  repeated  except  by  inver- 
sion as  4  +  5  and  5  +  4  ;  but  unlike  that  series 
it  is  entirely  promiscuous. 

To  make  a  series  for  subtraction,  write  the 
series    as    above,   and   write  the   sums  ot  the 
numbers  above,  thus  : 
10  14  II  15  12  16    7  II  15  12  16  13.   &c. 
69584736958473  &c. 
456789456789    &c. 
Then  copy  for  the  minuends  the  upper  num- 
bers, and  for  the  subtrahends  the  lower  ones 
and  the  series  becomes 

10  14  II  15  12  16    7  II  15  12  16  13    &c. 
456789456789   &c. 
In  this  book  those  for  subtraction  have  been 
still    further    disarranged    so    that  the  results 


8  PREFACE  TO  TEACHERS'  EDITION. 

will  not  be  6958473  and  so  on  lest  the  pupils 
notice  it  and  recite  that  instead  of  subtracting. 
For  division  find  the  products  for  the  upper 
line  instead  of  the  sums. 

For  division  with  remainders,  which  is  an 
excellent  preparation  for  short  division^  after 
having  written  the  products  above  as  before, 
add  10  each  one  of  them  a  number  less  than 
the  lower  number  in  that  column  and  write 
for  the  upper  line  these  sums  and  for  the 
lower  line  the  lowest  line  ot  figures. 

In  the  first  series  of  division  with  remain- 
ders, the  remainders  are  very  small,  that  it 
may  be  as  easy  as  possible. 

In  the  first  series  the  combinations  do  not 
go  as  far  as  9—  that  is  2  and  9,  9  times  2,  &c., 
but  only  to  combinations  of  2  3  4  5  and  6  with 
2345  and  6.  The  examples  which  follow 
immediately  after  the  learning  of  any  table 
involve  only  what  is  contained  in  the  table. 

It  is  believed  there  is  a  very  large  amount  of 
work  for  practice,  both  in  the  Pupils'  Edition 
and  in  thje  Teachers'  Edition,  more  than  twice 
as  much  as  in  other  works  of  the  kind.  The 
greatest  care  has  been  taken  that  they  may 
proceed  from  the  easiest  to  those  involving 
every  difficulty  which  pupils  should  meet  at 
the  age  for  which  this  book  is  designed. 

As  anything  is  learned  it  is  immediately 
put  into  use. 

LEVI  N.  BEEB& 

Canandaigua,  N.  Y.,  July,  1877. 


FIRST  STEPS  AMONG  FIGURES. 


PART    I. 

ONE. 
(See  Appendix,  page  185.) 

I.  Be  sure  the  pupil  has  the  idea  of  one  thing 
in  distinction  from  two  or  more  things. 

Illustrate  by  objects  as  much  as  possible, 
using  small  sticks,  or  square  blocks  \  inch 
square  and  \  inch  thick  of  different  colors,  or 
bright  cents.  Only  ten  of  each  are  needed  and 
if  the  teacher  has  ^// these  he  can  add  interest 
to  the  exercises. 

An  abacus,  or  numeral  frame,  is  almost  indis- 
pensable as  a  further  help. 

Show  the  pupil  that  taking  one  article  (as  a 
bean,  a  cent  or  a  block)  one  time,  or  putting  it 
into  a  box  or  upon  a  book  or  table  makes  one 
article  there,  which  is  the  interpretation  of 
"once  one  is  one."  Let  the  pupil  place  the 
article,  and  thus  impress  his  mind  more  thor- 
oughly with  the  idea  once  i  is  i,  written  i  x  i 

•    *In  this  book  the  multiplier  is  uniformly  placed  on  the 
right  of  the  sign   X  :  thus  z  times  one  are  2  will  be  written 

I   X  2  =  1. 

See  Appendix,  pp.  185-192. 


lO  FIRST  STEPS  AMONG  FIGURES. 

2.  The  idea  of  division  may  be  taught  in  the 
following  way  :    The  teacher  may  place  a  pile 
of  2  blocks  on  a  table  or  book  and  ask  the- 
pupil  •*  How  many  limes  one  block  have  I  in 
this  pile  ?  "     Pupil :  "  Two  times."     Teacher  : 
*'  One  block  in  two  blocks  how  many  times  ? " 
Pupil:  "Two  times."     Teacher:  "One  in  two 
how    many    times  .^ "     Pupil:    "Two    times." 
(The  pupil  may  use  in  each  answer  the  word 
"twice"  instead  of  the  words  "two  times.") 
Teach  that  this  is  written  2-^1  =  2  and  should 
be  read  by  the  youngest  pupils,  i  in  2  twice. 
*  Schedule: 
T  X  1  =  1.    (Read  once  one  is  one.) 
1-^-1  =  1.    (Read  one  in  one,  once.) 

What  can  you  find  once  in  the  school  room, 
in  your  pocket,  on  your  face,  at  home  t  &c. 

What  is  there  that  moves  on  one  wheel  ? 


TWO. 

3.  Teach  in  counting  that  the  second  of  two 
things  is  not  of  itself  Hvo,  hxiXmie. 

In  teaching  number  and  in  operations  on 
numbers  use  objects  for  some  time  —  at  least 

*Thc  schedules  being  written  on  the  blackboiard,  the  pupils 
are  to  be  taught  to  read  them,  and  eventually  to  make  them 
diemselves. 


FIRST  STEPS  AMONG  FIGURES.  1 1 

three  months  to  six  months,  until  the  pupil  is 
thoroughly  familiar  with  the  composition  of 
numbers. 

4.  Teach  pupils  to  count  to  ii  and  continue 
to  teach  counting  daily  until  the  pupil  can  count 
100. 

Schedule : 
1  +  1  =  2.  (Read  one  and  one  are  two.) 

1  X  2  =  2.  (Read  twice  one  are  two.) 

2  —  1  =  1.  (Read  one  from  two  leaves  one.) 
2  —  1  =  2.  (Read  one  in  two  twice.) 

2  is  one  nrjore  than  what  number } 

1  is  one  less  than  what  number .? 

2  is  the  double  of  what  number? 
2  is  twice  what  number? 

I  is  one-half  of  what  number? 

I  and  I  are*  ?     i  from  2  leaves  ? 

(i  from  2  leaves  i,  because  i  and  i  are  2.) 

Mary  has  2  sticks  of  candy  ;  she  gives  away 
2  sticks  ;  how  many  sticks  has  she  left  ?  2  from 
2  leaves  ? 

Henry  had  2  marbles  ;  he  gives  none  away  : 
how  many  has  he  ?     Nothing  from  2  leaves  ? 

•The  teacher  will  supply  the  words  "  what  Qumber  '* 
or  "  how  much  "  in  such  examples,  according  to  the 
sense. 

Show  the  pupils  that  i  block  placed  on  the  table,  and 
then  another,  make  2  blocks  there,  hence  2  is  2  times  I. 


12  FIRST  STEPS  AMONG  FFGURES. 

What  is  there  that  moves  on  two  wheels  ? 

Hold  up  two  fingers. 

What  have  you  on  your  head  of  which  there 
are  2  and  only  2  ?  In  the  school  room  ?  At 
home  ?  &c. 

What  animals  walk  on  2  legs  ? 

5.  What  is  i  of  an  apple?  (Let  the  pupil 
take  an  apple  and  cut  it  into  halves  and  ask 
him  what  one  piece  is  called.  Show  him  that 
if  he  takes  one-half  of  the  apple  there  is  left 
as  much  as  he  takes.)  What  is  i  of  2  apples  ? 
Placing  2  apples  on  the  table,  let  one  of  the 
pupils  take  half  of  them  by  leaving  as  much  as 
he  takes. 

(Vary  the  exercise  by  taking  J  of  a  stick  of 
candy,  *  of  2  sticks,  &c  ) 

6.  2  is  the  double  of  what  number?  Of  what 
number  is  i  one-half? 

What  number  must  I  double  to  get  2  ? 
I  know  a  number  that  is  i    more  than  i  ; 
what  number  is  it  ? 

7.  What  number  must  be  added  to  i  to  get 
2  ?  Fred  had  2  dimes  and  bought  peaches  with 
I  dime.     How  many  dimes  had  he  left  ? 

(No  analysis  of  these  examples  is  expected; 
simply  a  prompt  answer.) 


FIRST  STEPS  AMONG  FIGURES. 


A  slate  pencil  costs  i  cent,  how  much  will  2 
slate  pencils  cost  ? 

Charles*  had  a  marble,  and  his  sister  had 
twice  as  many.     How  many  did  she  have  ? 

How  many  slate  pencils  can  you  buy  for  2 
cents  ? 

How  many  2-cent  stamps  can  you  buy  for  2 
cents?     How  many  i-cent  stamps? 

(Both  these  and  the  following  examples 
should  be  gone  over  many  times,  taking  them 
in  a  different  order  each  time  and  often  giving 
them  promiscuously.) 

8.  Teach  that  there  are  2  pints  in  a  quart  by 
pouring  a  pint  cup  full  of  water  twice  into  a 
quart  cup. 

What  cost  a  quart  of  milk  at  i  cent  a  pint? 


THREE. 

Schedule : 
9.  Measuring  by  i. 
I    I    I     3. 
/  I     i  +  i -1-1=3. 

^i     iX3=3-  [=1. 

/j_    3  — 1-1  =  1,  for  3  — 1  =  2  and  2  — I 

s  3-^1=3. 


♦In  such  examples  it  will  interest  the  class  to  use  tkti* 
names  instead  of  those  given. 


14  FIRST  STEPS  AMONG  FIGURES. 

Measuring  by  2. 
I   I  2     2  +  1=3. 

II     2  X  I  +  I  =3.    (To  be  read  once  2 
III  3  and  1  are  3,  or  once  2  plus  i 

are  3.) 
3—2  =  1,3  —  1  =  2,    (To  be  read  2 
from  3  leaves  i  and  i  from  3 
leaves  2.) 
3-1-2  =  1    (and    I  rem.)     (To   be 
read  2  in  3  once  and  i  re- 
mainder.) 
The  pupils  should  read  these  schedules  many 
times  each,  until  they  are  familiar  with  the  lan- 
guage. 

10.  Illustrate  by  a  pile  of  3  blocks.  How 
many  times  have  I  2  blocks  in  the  pile  .^  Once. 
Take  them  away  once  then.  How  many  are 
left,  or  how  many  remain  ?  One.  2  blocks  in 
3  blocks  how  many  times?  Once  and  i  re- 
mainder. 2  in  3  how  many  times }  Once  and 
I  remainder. 

11.  To  be  written  on  the  blackboard  for 
pupils  to  bring  written  with  the  answers  to  reci- 
tation. 

3-i-i=?  iX2=?  3  —  1  +  1=? 

I-fl=?  1+2=  ?  2|2=? 


FIRST  STEPS  AMONG  FIGURES.  15 


3  —  2=?  3-1=?  2  —  2=? 

2+I=?  2+1+1=?         IX2+I=? 

2-J-2=?  3^2=?  3-^3=? 

12.  3  is  I  more  than? 

1  is  I  less  than  ? 
3  is  2  more  than  ? 

2  is  I  less  than  ? 

2  is  I  more  than  ? 
I  is  2  less  than  ? 

3  is  3  times  ? 

13.  To  illustrate  tell  a  pupil  to  put  one  block 
or  one  cent  on  the  desk  and  then  another. 
Show  the  pupils  that  a  block  has  been  put 
upon  the  desk  twice  and  that  there  are  two 
blocks  there  ;  hence  2  times  i  block  are  2 
blocks  ;  also  2  times  i  orange  are  2  oranges, 
and  2  times  i  pencil  are  2  pencils,  &c.  2  times 
any  one  thing  are  two  of  those  things.  2  times 
I  are  2. 

Show  the  pupils  that  i  block  taken  3  times 
or  placed  on  a  table  3  times  makes  3  blocks 
there,  hence  3  is  3  times  i. 

14.  This  form  of  illustration  may  be  used  foi 
any  multiplication. 

How  many  pints  in  a  quart? 

Teach  pupils  to  write  numbers  as  high  as  20. 


l6  FIRST  STEPS  AMONG  FIGURES. 

It  may  be  well  to  teach  the  writing  of  12  before 
10  or  II.  Show  the  pupil  by  the  abacus  or 
otherwise  12  objects  and  show  him  that  they 
are  i  ten  and  2  ones.  Show  him  that  we  can- 
not write  12  by  any  one  of  our  figures  ;  then 
teach  him  about  ten's  place  and  one's  place. 

Do  not  use  the  word  units  for  several  weeks 
yet. 

15.  Teach  pupils  to  count  by  2's  from  2  to  6 
and  back  to  2,  thus  :  2,  4,  6.     6,  4,  2. 

Explain  that  i  and  i  are  equal  numbers,  that 
is  equal  to  each  other  ;  i  and  2  as  well  as  2  and 
3  are  unequal  numbers. 

16.  Give  the  pupils  much  practice  in  exam- 
ples like  the  following:  3  — i  — i  +  i?  To  be 
read,  how  many  are  3  less  i  less  i  plus  i  ?  or 
3,  subtract  i,  subtract  i,  add  i  ;  or  3  minus  i 
minus  i  plus  i.* 

*Thesc  arc  to  be  read  by  the  teacher,  thus:  3,  add  a» 
subtract  1,  divide  by  1,  multiply  by  3. 

The  examples  may  be  read  through  and  those  who  can 
answer  raise  the  hand  \  the  teacher  call  upon  one  most 
unlikely  to  be  correct  for  the  answer  j  if  incorrect  call  upon 
another  until  the  correct  answer  be  given. 

It  may  be  best  at  first  and  perhaps  often  to  have  the  result 
of  the  first  step  given  by  one  pupil,  the  next  step  by  the  next> 
&c     E.  g.  teacher,  3,  add  2.      1st  pupil  says  •'  5."    Teacher, 


FIRST  STEPS  AMONG  FIGURES.  17 

2—1+2  —  1X1=?  1-1-2  —  1  —  1X2  +  1=1 

2+1—2X3—1+1=?     1+1X1—1+2—1=1 

Read  with  as  much  rapidity  as  the  class  can 
follow  silently  and  give  the  answer  at  the  end, 
the  rapidity  being  increased  as  the  pupils  have 
more  practice. 

17.  From  what  number  can  you  take  one 
and  have  one  left  ? 

Count  by  2's  from  2  to  10. 
What  number  is  twice  i  1 

18.  I  write  a  number  once,  and  again,  to  get 
2  ;  what  number  did  I  write  twice  ? 

How  many  cents  must  you  have  to  buy  a 
3cent  stamp? 

Mary  had  to  get  a  pound  of  tea  for  $1  ;  her 
mother  gave  her  $3  ;  how  much  money  ought 
she  to  bring  back  ? 

Henry  learned  i  line  in  his  primer,  and  his 

"subtract  1."  ad  pupil  says  "4."  Teacher,  "divide  by  2."  3d 
pupil  "  a."  Teacher,  "  multiply  by  3."  4th  pupil  "  6."  Call 
on  an  inattentive  pupil  at  any  step  in  these  examples  for  the 
answer.  Usually,  the  teacher  reads  the  whole  example  and 
the  pupils  give  only  the  final  answer. 

The  foregoing  examples  are  not  written  so  as  to  be  cor- 
rect for  solving  from  the  written  or  printed  form  for  in  that 
case  3  —  2  X  *  would  mean,  take  2  x  *  from   3,  but  it  is  to 
be  read  :    3  subtract  2,  multiply  by  2. 
2 


1 8  FIRST  STEPS  AMONG  FIGURES. 

sister  learned  i  line  more  than  he  did ;  how 
many  did  she  learn? 

If  I  slate  pencil  cost  i  cent  what  will  3  slate 
pencils  cost  ? 

Anna  found  3  roses  in  the  garden;  how  can 
she  divide  them  between  her  father  and  mother  1 

Can  she  give  them  an  equal  number  ? 

How  maqy  roses  must  she  have  had  in  order 
to  give  her  father  i,  and  her  mother  i  also  ? 

Count  by  2's  from  from  2  to  12. 

The  counting  may  be  sometimes  in  concert, 
oftener  ist  pupil  say  2  ;  2d  pupil,  4  ;  3d,  6  and 
so  on,  and  perhaps  oftenest  one  pupil  give  the 
whole  series. 

Teach  pupils  to  count  ist,  2d,  3d,  &c. 


FOUR. 

20.  Measuring  by  i. 

I   I   I   I     4. 
/i     14-1  +  1  +  1=4.      (Because   i  +  i=a, 

2  +  i=3»  3  +  1=4-) 
/  I     1  X4=4. 

/i     4  — I  — I  — 1  =  1, or  4— I  — I  — I  — 1=0. 
/  i_   4-M=4. 
S  4 


FIRST  STEPS  AMONG  FIGURES. 


19 


Measuring  by  2. 
112.     2  +  2=4. 
2  X  2=4. 
112.     4  — 2  =  2,  or  4  —  2— 2=a 
I  III     4.     4-^2  =  2. 
Measuring  by  3. 
Ill     3.     3  +  1=4,  1+3=4- 
3  +  1  +  1=5- 

!_  i:     4-3  =  1^4-1=3- 

nil  4.     4-+3  =  i  (and  i  remainder.) 

21. 

2  X  I  +  2=  ? 


3  —  2=? 
4x1  = 

3-1  = 
2  +  2  = 
3-^2  = 
4-2  = 
1x3  = 

4-7-2  = 


3-i=? 

2  +  I=? 
2X2=? 


4-^3  = 
3-^2  = 

3  +  1  = 
4-3  = 
4-1  = 
4-4=? 


2  X 1  +  1=  ? 
3x1+1=? 
4  —  2  —  2=  ? 
4-f-4=? 

22.  Name  animals  with  4  legs  ;  with  2  legs. 
Name  wagons  and  vehicles  with  i  wheel  ;  2 

wheels  ;  3  wheels ;  4  wheels.  Compare  them. 
(For  instance  a  wagon  with  4  wheels  has  how 
many  more  wheels  than  one  with  2  wheels  ?  &c.) 

23.  4  is  I  more  than  ? 
I  is  I  less  than  ? 


20  FIRST  STEPS  AMONG  FIGURES. 

2  is  I  more  than  ? 

3  is  I  less  than  ? 

I  is  I  more  than  ? 

1  is  2  less  than? 

2  is  I  less  than  ? 
2  is  2  more  than  ? 
2  is  2  less  than? 

4  is  2  more  than  ? 

I  is  I  more  than  ?  (Nothing.) 
4  is  4  times  ? 

24.  Solve  rapidly  the  following: 
2x2—3+2x1+1—2x2=? 

4— I  — I  +  I  +  I— 3  =  how  many  less  than  4? 
3"-2  +  3— I  — I  X  2—  i=how  many  times  i  ? 
I  +  2  —  1-^2  +  2-1  =  how  many  more  than  2  ? 
Teach  to  count  by  2's  from  2  to  20  and  back 
to  2. 

25.  1x2  — I  X  3  —  2  — 2  =  how  many   less 
Ihan  3  ? 

3—2+1 X2— 1—2+1= ? 
4—2—1 X3— 1X2— I=? 

26.  What  number  must  I  double  to  get  4? 
Of  what  number  is  4  the  double  ? 

Of  what  number  is  2  one-half? 

What  number  can  be  taken  twice  from  4? 

W^hat  number  is  2  more  than  i  ? 

What  number  must  I  add  to  2  to  get  4? 


FIRST  STEPS  AMONG  FIGURES.  21 

"What  number  is  1-2  of  four? 

How  many  less  than  3  is  the  half  of  4  ? 

27.  Minnie  had  4  pinks  which  she  neglected 
sadly  ;  one  day  i  of  ihem  withered,  the  second 
<iay  another,  and  the  following  day  i  more. 
How  many  fresh  ones  had  she  then  ? 

How  many  $*s  are  $2  +  $2  ? 

How  many  apples  are  3  apples  and  2  apples 'i 

28.  Teach  that  there  are  4  quarts  in  i  gal- 
^on,  taking  a  gallon  measure  and  filling  it  by 
ipouring  a  quart  measure  full  of  water  into  it  4 
times. 

Nellie  bought  a  gallon  of  milk ;  how  many 
quarts  did  she  buy  ? 

She  paid  i  dime  for  each  quart ;  how  many 
dimes  did  she  pay  for  the  gallon  ? 

If  2  qts.  of  milk  cost  2  di.,  can  you  get  a 
gal.  for  3  di.  ? 

How  much  can  you  get  for  the  3  di.  ? 

If  I  drink  a  quart  of  milk  in  2  days,  what 
part  of  a  qt,  do  I  drink  in  i  da.  ? 

29.  William  having  4  apples,  ate  half  of  them 
iind  one  more,  how  many  had  he  left  ? 

What  number  is  1  more  than  half  of  4  ? 

Ann  had  3  apples ;  she  gave  an  equal  number 
to  her  mother,  father  and  brother;  how  many 
did  she  give  each  ? 


22 


FIRST  STEPS  AMONG  FIGURES. 


Sarah  cut  i  apple  into  2  equal  pieces;  what 
would  you  call  one  of  the  pieces  ? 

Teach  to  count  by  2's  from  2  to  40  and  back 
to  2. 

By  marking  off  paper  or  pasteboard,  or  bet- 
ter a  thin  board,  and  cutting,  according  to  the 
following  directions,  an  excellent  aid  in  teach- 
ing notation  and  numeration  may  be  obtained. 
By  ruling  both  ways,  mark  off  into  squares  10 
squares  in  a  row  and  21  rows,  as  shown  below. 
Cut  off  one  row  or  strip  of  10  squares  and  theft 
cut  up  the  strip  into  single  squares.  After- 
wards cut  off  10  strips  of  10  squares,  which 
will  leave  a  large  square  containing  10  rows  of 
small  squares  with  10  squares  in  each  row. 


The  10  small  squares  cut  up  may  be  used  to 
illustrate  ones  or  units,  and  the  strips,  tens. 


FIRST  STEPS  AMONG  FIGURES.  2$ 


while  the  large  square  will  represent  hundreds. 
With  them  it  will  be  easy  to  show  that  lo  ones 
equal  a  ten,  and  ten  tens  one  hundred,  and  the 
teacher  will  show  where  the  ones,  tens,  and 
hundreds  are  written  in  numbers. 


FIVE. 

Schedule; 

30.  Measuring  by  i. 

I 

I    I    I    I     5. 

/  1      1  +  I  +  I  ♦- 1  +  I  =5. 

/  1      ly 

^5  =  5- 

/I     5- 

-1  —  1  —  1  —  1  =  1. 

/i     5^ 

-1=5. 

/  I 

^5 

Measuring 

by  2. 

I    1      2. 

2  +  2  +  1=5. 

1    I     2. 

2x2  +  1=5.     (See  note  p.  23.) 

I     2. 

5-2-2=1. 

mil  5- 

5-i-2  =  2  (i  remainder.) 

By3- 

I    I    I     3. 

3  +  2  =  5.2+3  =  5- 

1    1     2. 

3x  1+2  =  5. 

inn   5. 

5-3  =  2- 

5  +  3  -I  (2  rem.) 

24  FIRST  STEPS  AMONG  FIGURES. 


By  4. 

I   I   I   I     4.     4  +  1=5,  I- 

+  4=5* 

I     I.     4x1  +  1=5. 

mil         5.     5-4=»- 

5^4=1  (I 

rem.) 

3'. 

3  —  1  =  ?         1x3  +  1=? 

5-2-2=1 

5  +  2=?        2x2  +  1=? 

5x1=? 

4-2=?        5-2=? 

5-5=? 

2x2=?        3+i+i=? 

5-4=? 

4-^3=?         4-2-2=? 

3X I+2=? 

5-3=?        3-3=? 

4+-i=? 

4+i=?         5-4=? 

2+2  +  1=1 

5^3=?         2+2=? 

32.  5  is  I  more  than  ? 

2  is  I  less  than  ? 

3  is  2  less  than  ? 

4  is  2  more  than  ? 

VVhat  number  added  to 

2  will  make  5  ? 

5  IS  4  more  than  ? 

3  is  I  more  than  ? 

5  is  2  more  than  ? 

2  is  2  less  than  ? 

3  is  3  more  than? 

I  is  2  less  than  ? 

5  is  how  many  times  i  ? 

FIRST  STEPS  AMONG  FIGURES.  25 

(Since  2  is  added  iwice  and  i  is  added  to  the 
result  to  get  5,  2  times  24-1=5  and  since  2  may 
be  subtracted  from  5  twice  and  i  will  remain, 
2  is  contained  in  5  twice  and  i  remainder,  or 
5  +  2  =  2  [i  rem.].) 

33.  Teach  pupils  to  write  and  read  Roman 
notation  to  V.  Teach  pupils  to  count  by  a's 
from  2  to  50, and  back  to  2,  and  from  i  to  ii 
and  back. 

34.  For  rapid  solving. 

5  —  2  —  3  +  2x2—3  +  2=?  Ans.  3. 
2x2  +  1—3x1x2—3+3=?  Ans.  4. 
4—1+2—3x2  —  1  —  2x3  —  1=?  Ans.  2. 
3  +  2  —  1  —  2x2—3—1=?  Ans.  o. 
5-3  +  2-3x3-1+2=?  Ans.  4. 
2+2—3x2+3  —  2  +  1  —  2=?  Ans.  2. 

2  + 1—2  X  3—3 +  2  X  2  is  how  much  more 

than  I    ?  Ans.  3  more. 

3  +  2—1  —  2  +  1  —  2x3  —  1=?     Ans.  2. 
Review  these  frequently. 

35.  Review  counting  by  2's. 

Teach  to  count  by  2's,  commencing  with  i, 
to  21. 

36.  How  many  must  I  add  to  3  to  get  5  ? 
How  many  must  be  taken  from  5  to  get  3  ? 
Why?     (Because  2  added  to  3  makes  5  ) 


26  FIRST  STEPS  AMONG  FIGURES. 

37.  How  many  times  2  must  I  add  to  i  to  get 

s? 

I  have  taken  away  twice  i  from  a  certain 
number  and  2  remains.    What  number  was  it? 

I  have  taken  2  from  a  certain  number  and  r 
remains.    What  number  was  it. ^ 

I  have  added  2  to  a  certain  number  and  have 
3.     What  number  was  it? 

How  many  gallons  are  2  quarts.    Ans.  None. 

John  had  5  dimes  ;  he  bought  2  copy  books, 
each  of  which  cost  2  dimes.  How  many  dimes 
did  he  keep.^     (Illustrate,  using  dimes.) 

George  read  a  lesson  once.  Helen  read  it  as 
many  times  as  he  did  and  two  times  more. 
How  many  times  did  she  read  it? 

A  father  had  5  peaches  and  gave  them  to  his 
3  children  ;  he  gave  the  oldest  i  peach,  and 
gave  to  each  of  the  others  an  equal  number  ; 
how  many  did  each  of  the  younger  children 
receive  ? 

A  boy  has  2  cents,  he  finds  2  cents ;  how 
many  will  he  have  to  earn  to  have  5  cents? 

1  is  J  of  what  number  ? 

James  has  5  marbles,  he  loses  2  ;  how  many 
more  than  2  has  he  left  ? 

2  boys  are  passing  my  house,  and  each  boy 


FIRST  STEPS  AMONG  FIGURES.  2f 

is  driving  2  goats  ;  how  many  goats  are  pass- 
ing my  house  ? 

David  rode  i  horse  from  the  pasture  to  the 
barn  and  at  the  same  time  led  2  others  ;  how 
many  horses  did  he  bring  to  the  barn  ? 

Jane  had  5  chickens.  A  rat  ate  i  of  them^ 
and  then  a  cat  ate  half  of  what  were  left  and 
I  more  ;  how  many  lived  ? 

A  boy,  having  4  pockets,  has  2  apples  in  i  of 
them  ;  one  pocket  is  empty,  and  he  has  i  apple 
in  each  of  the  other  pockets  :  how  many  apples^ 
has  he? 

38.  Teach  counting  by  2's  from  i  to  21  and 
back.  Review  former  counting.  (Do  not 
teach  all  ot  this  before  giving  other  exercises^ 
but  require  some  of  this  kind  of  exercise  daily 
until  as  much  as  is  denoted  above  has  been  ac- 
complished. These  directions  apply  to  future 
countings.) 


SIX. 

Schedule : 

39.  Measuring  by  i.   (Teach  pupils  to  make 
ihese  schedules.) 

I    I    I    t    I    I.     6. 


a8  FIRST  STEPS  AMONG  FIGURES. 

/  I     1  +  1  +  1  +  1  +  1  +  1=6. 
/   I     I  x6  =  6. 
/   I     6—1  —  1  —  1  —  1  —  1=1. 
/  I     6^1=6. 
/  I 
/_! 
6  6 
By  2. 

I      I       2       2+2+2=6. 

I      I        2        2X3=6. 

112       6  —  2—2  =  2,   6  — 2  — 2— a=SOt 

I  I  I  I  I  I    6     6-^-2=3. 

By  3 

I    I    I     3     3+3=6. 

I    I    I     Z_    3x2=6,2x3=6. 

I  I  I  I  I  I     6     6-3  =  3,  6-3-3=a 

6-3  =  2. 

By  4 

J    I    I    I     4.     4  +  2=6. 

II     2.     4x1  +  2=6. 
I  I  I  I  I  I      6.     6—4=2,  6  —  2^4. 
6-r-4=i  (2  rem.) 
By  5. 

I    I    I    I    I    5.    5  +  1=6,  I  +5=6. 
I     I.    5x1  + 1=6. 
I    I    I    I    I    I     ~6.     6  —  5  =  1,6  —  1=5,6  — 

5-1=0. 
6-^5  =  1  (i  rem.) 


FIRST  STEPS  AMONG  FIGURES. 


29 


40. 

2  X  2  +  I  =  ? 
1  +  1+4=  ? 
6+2=? 


5  —  2=.''  2X2  +  1=.''         4—2=? 

3+3=?  i+i+4=?         i+2+l=? 

2X2=?  6+2=?  6— 2  — 2— 2—? 

I+4+I?     3+2=?  4+i=? 

5-+I=?  5—1  —  1==?         2X2  +  2=? 

3x1=?       6-2=?  5-3=? 

6— 2  — 2  =  ?1X5=?  4— 1  — 1  — 1«— ? 

41.  6  is  3  more  than  ? 
What  is  J  of  6  ? 

6  is  4  + how  many  ? 

6  is  2  times  what  number? 

How  many  times  can  you  take  4  from  6? 

6  is  3  times  what  number? 

4  less  than  6  is  ? 

6  is  2  +  ? 

What  number  is  3  less  than  6? 

What  number  is  half  of  4? 

6  is  1  +  ? 

42.  For  rapid  solving. 
1+2+2=?     3+2—1+2^1 

I+1+I42=?       2+2+2=1 
3  +  2  —  1+2  X3=  ? 

5  —  2+3  —  2  —  3  +  1x2  +  1—3  +  1'?  Ans.  3. 
3  +  1  —  2x3-3  +  1-2  +  1x2—2  +  2—? 

Ans.  2. 


30  FIRST  STEPS  AMONG  FIGURES. 

2+"3— 2X2  — 5x34-2— 3X1  +  2?  Ans.  4. 
4—3  +  2—1  X2-f  2  +  2-1-2-1-2-1-2  +  2? 

Ans.  16. 
5  +  2  +  2+2  +  2  +  2  +  2  +  2  +  2  +  2  +  2  +  2? 

Ans.  27. 

43.  Count  by  3's  from  3  to  12.  Review 
■counting. 

Co.int  by  3's  from  3  to  18.  Review  the 
counting  already  taught.  Teach  Roman  nota- 
tion to  X. 

44.  For  rapid  solving. 

6  —  2  —  2+3  —  2X2+2-1-2   1  Ans.  10. 
2x3— 4  +  3—  I +2  +  2  +  2+2+2  +  2+2  1 
Ans.  18. 

5—3  +  1  +3-2-^-2+3  +  2  +  2  +  2+2+2 

+  2  +  2  1  Ans.  19. 

1^.3  —  2+3  +  1-^3  +  2  +  2  +  2+2+2  =1 

Ans.  12. 

5  —  2  —  2x2  —  1+4  —  3x3+3+3+2+2 

+  2=  1  Ans.  18. 
a +3  — 2  +  3-+2  +  2  +  2  +  2  +  2  +  2+2  + 

2=  ?     Ans.   17. 
4—3  +  2  +  2—3x2  +  2  +  2+?     Ans.  8, 
6-^3+3-2  X  2+3  +  3  +3  +  2  +2+2 

+  2/     Ans.  23. 

45.  Examples  in  addition  may  be  written  on 


FIRST  STEPS  AMONG  FIGURES.  3 1 

the  board,  a  few  for  the  pupils  to  solve  each 
day  on  their  slates  and  bring  to  class ;  and  ad- 
Aiitional  examples  may  be  read  to  the  class  and 
solved  on  the  slates  or  on  the  board.* 

46.  3  cts.  is  2  cts.  more  than  John  has ;  how 
many  has  he? 

3  cts.  is  1-2  of  what  money  Jane  has;  how 
much  has  she  ? 

2  cts.  ia  3  cts.  less  than  an  orange  cost ;  how 
many  cts.  did  it  cost  ? 

I  is  how  many  less  than  6  ? 

6  cts.  will  buy  how  many  3  cent,  stamps  ?  1 
•ct.  stamps  ?     2  ct.  stamps  ?     5  ct.  stamps  ? 

George  had  1-2  of  6  cents;  how  many  had 
he? 

Clara  had  6  flowers;  she  gave  them  to  her 
father  and  mother.  If  she  gave  each  of  them 
the  same  number,  how  many  did  her  father 
get  ?     Her  mother  ? 

Charles  had  6  cts. ;  he  lost  2  of  them ;  how 
many  had  he  left  ? 

Carrie  had  6  peaches :  she  gave  her  father  2 
of  them  and  her  mother  2  ;  how  many  did  she 
keep  ? 

3  cts.  is  3  cts.  more  than  Byron's  money ; 
iiow  much  money  has  he  ? 

•  See  T.  £.d.,  p.  84,  and  P.  Ed.,  p.  33. 


32  FIRST  STEPS  AMONG  FIGURES. 

47.  3  +  2  —  1+2  —  5+3=?     Ans.  4. 

1-2  of  6  +  2  — 3  is  what  part  of  4?  Ans.  1-2, 

1+2  +  2  —  3—1+3—2=?     Ans.  2. 

3  +  2-1-3  +  1x3  +  3  +  3  +  3  +  2  +  2=; 

Ans.  19. 
6  —  4  +  2—  1x2  —  2  +  2  +  3+3  +  2  +  2  + 
2?     Ans.  18. 
I  put  down  a  number  once  and  again  and 
again  to  get  6.     What  is  the  number  ? 

48.  Count  by  3's  from  3  to  24.  Review 
the  counting. 

From  what  number  can  you  take  2x2  and 
keep  1  ? 

What  number  must  I  double  to  get  4  ? 

What  number  is  one  less  than  5  ? 

What  is  1-2  of  4  ? 

What  is  ^  of  the  number  i  less  than  5  ? 

What  number  is  one  less  than  i^  of  4  ? 

Augusta  had  5  cents.  She  lost  1  of  them^ 
and  spent  ^  of  what  she  had  left,  and  then 
found  3  cents.  How  many  had  she  then  ? 
(Solve  one  step  at  a  time.) 

49.  For  the  method  of  making  series  like  the 
following,  see  preface.  For  method  of  use  see 
pages  42  and  43. 

For  addition.* 

*  Teach  each  series  thoroughly  before  taking  the  next. 


FIRST  STEPS  AMONG  FIGURES. 


33 


a 
420 


b 

^  3 
2   I 


I   2   I 
For  subtraction. 


c 

2  o 
I   2 


d 

I  3 
I   2 


a 

b 

c 

d 

e 

f 

5  2 
3   I 

5  3 
2   I 

6  2 

2  2 

5  4 
I   2 

3  I 

2   I 

6  3  4 
3  3  1 

For  multiplication. 


b 
3  1 


2  o 
I  2 

For  division, 
a        b 
6230 

3  2 


o   I 


I   2 


c 
2  o 

2    O 

C 
2  3 
I  3 


d 

0  6 

1  2 


For  addition.     (Re-arranged.) 


a 
4  3 
'  3 


b 

2    0 
2     I 

C 

2  I 

3  2 

d 
3  0 
I  3 

e 
4  2 
2   I 

f 
0  2 
2  3 

g 

I  3 
I  2 

For  subtraction. 

(Re- arranged.) 

a 

b 

c 

d 

e 

f 

g 

4  1 

5  6 

3  4 

3  3 

2  5 

2  5 

6  5 

2  I 

3  2 

3  I 

2   I 

2  3 

I  2 

3  I 

50.  Teach  Roman  notation  to  XVIII. 

Review  counting  by  2's  commencing  with  2, 
and  also  with  i,  to  60  and  61. 

Teach  counting  by  3's  commencing  with  3 
to  30. 
3 


34  FIRST  STEPS  AMONG  FIGURES. 

The  examples  given  under  previous  numbers 
should  be  frequently  reviewed,  so  that  the  pupil 
may  become  quite  familiar  with  the  formation 
and  use  of  numbers. 


SEVEN. 

51.  At  this  Stage  pupils  should  make  the 
schedule  from  their  memory  on  the  plan  of 
those  already  given. 

Schedule: 

Measuring  by  i 

iiiiiii     7. 

1  +  14- 1  +  1  + I +  1  +  1=7. 

1x7.     7. 

7  — I  — I  — I  — I  — I  — 1=1,  or  7—1 — I — I 

I  —1  —  1  —  1=0. 
7-M=7. 

By  2. 

112.     2  +  2  +  2 +  1—7. 
I    I    2.     2x3  +  1=7. 

112.       7  — 2  — 2  — 2««I. 
I      I. 


7  7-      7^2=3  (i  rem.) 
Teach  pupils  to  write  Arabic  to  lOO. 


FIRST  STEPS  AMONG  FIGURES.  35 

By  3- 

1113      3  +  3  +  1—7- 
I    I    I    3.     3x2  +  1—7. 

I    I-     7-3-3—1. 

I  I  I  I  I  I  I    7. 

7-3=2  (i  rem.) 
By  4. 

I    I    I    I    4.    4  +  3=.7- 
1113'     4Xi+3='7- 

III  III  I     7.      7-4=3. 

7-^4=1  (3  rem.) 
By  5. 

I    I    I    I    I     5.    54-2=7. 

r    I     2,     5x1-1-2—7. 

I   I   I    I   I   I    I     7.     7-5-.2. 

7^5=1  (2  rem.) 
By  6. 

I   I   I   I   I   I     6.     6  +  i=-7. 

I      t.     6x1  +  1—7. 

I   I  I  I  I   I   I     7.     7_6=.i. 

7-^6=1  (i  rem.) 

5  +  i«?        7-5- .>  6-r-4-? 

6-2—?       1x5  +  1—?       7-3—? 


36  FIRST  STEPS  AMONG  FIGURES. 

5-M— ?  3X2  +  1—?  2X2  +  2—? 

3X2—?         6  —  2  —  2—?  7  —  2  —  2  —  2—? 

3  +  2—  ?      2X2—?  1x5  +  2—? 

4-2-?       7  +  4-?  5-J— ? 

6+1—?       3>^2— ?  7-^3—? 

2X2+  I—?  2  X  2+3—? 

53.  7  is  3  more  than  ? 

What  is  i  of  the  number  i  less  than  .7  ? 
7  is  I  more  than  twice  what  number? 
What  number  is  3  less  than  7  ? 
What  number  must  be  added  to  3  to  get  7  f 
How  many  times  can  you  subtract  2  from  7  ? 
7  is  4+  ?     3  is  J  of? 

I  added  3  to  a  certain  number  and  got  5  ; 
what  number  was  it  ? 

1  pear  is  what  part  of  7  pears  ? 

2  pears  are  how  many  times  i  pear  ? 

How  much  must  be  taken  from  7  to  leave  3  ? 
What  number  must  be  added  to  2  to  get  7  ? 
How  many  times  3  must  I  add  to  i  to  get  7  ? 
Count  by  3's  from  3  to  42.  Review  the 
counting. 

How  many  times  can  you  subtract  5  from  7  ? 

54.  6  is  double  what  number  ?  3  is  2  less  than  ? 
What  number  is  one  less  than  7  ? 

What  is  ys  of  the  number  i  less  than  7  ? 


FIRST  STEPS  AMONG  FIGURES. 


37 


55- 


For  addition. 


a 

b 

c 

d 

e 

f 

3M 

203 

120 

3'2 

0131 

420 

123 

412 

312 

342 

3441 

234 

For  subtraction. 


a 

b 

c 

d 

e 

f 

456 

277 

61S 

454 

353 

2643 

432 

143 

412 

312 

341 

2312 

b 

c 

d 

e 

03 

102 

10 

22 

32 

132 

12 

31 

For  multiplication. 

a 

31 
12 

For  division. 

a 

26 

13 

Same  re-arranged.  * 
For  addition. 


b 

c 

d 

e 

01 

40 

36 

023 

21 

21 

32 

321 

a 

b 

c 

d 

e 

f 

241 

302 

413 

021 

302 

4130 

"3 

412 

341 

231 

234 

1234 

For  subtraction. 


a 

b 

c 

d 

e 

f 

435 

256 

534 

634 

741 

6375 

432 

143 

412 

312 

341 

2312 

For  multiplication. 


a 

b 

c 

d 

e 

20 

12 

03 

12 

031 

12 

12 

31 

23 

"3 

For 


3» 


FIRST  STEPS  AMONG  FIGURES. 


For  division. 

a 

b      c 

d 

e 

32 

60    40 

63 

201 

13 

3J    23 

23 

131 

56.  After  earning  3  cts.,  Fanny  had  5  cts.  ; 
how  many  had  she  before  ? 

How  many  must  you  add  to  3  to  make  7  ? 

Nellie  has  2  pencils,  and  Sarah  has  i  more 
than  Nellie  ;  how  many  have  both  of  them  1 

Marcus  has  4  marbles  and  Arthur  has  2  less  ; 
how  many  have  both  ? 

Mary  has  6  pins  and  Stella  has  2  ;  how  many 
more  has  Mary  than  Stella  ? 

57.  When  the  pupils  become  listless  or  rest- 
less, or  a  minute  or  two  of  spare  time  is  at  com- 
mand, the  following  examples  and  like  exam- 
ples given  through  the  book  will  be  found  both 
useful  and  interesting.     Use  them  often. 

3-f2  +  2— 3  +  2H-2  — iX2-f3  — 2  — 3  +  2/ 

Ans.  4. 
5— 3  +  1  X  2 -3 +  2-3  4-4— 2x2  —4x3 

-fi?     Ans.  7. 
4  +  3-2 -3 -fiX2 -3 +  2  — 3x3-1-2+4 

—  2  ?     Ans.  5. 

7  —  2-3  +  4-2  +  3—5  +1  X2— 3  +  3— 4 

—  2  ?     Ans.  o. 


FIRST  STEPS  AMONG  FIGURES.  39 

2-1-5— 3-i-2  — 2f3  X2  — 2^2  X  3^2  —  1  ? 

Ans.  2. 
5-2  — 2  X  7-5  X  2-3 -M  X  3— 4  +  3  — 2 -H 
4-4?     Ans.  3. 

6— 4  +  3 -I- 2  —  I -=-3  X  2 -h  3 — 4  +  2  +  2 +3  + 
3  +  3  ?     Ans.  16. 

59.  Teach  the  pupils  to  read  numbers  to  looi. 
Show  them  that  the  figure  in  the  third  place 
represents  hundreds  (or  is  named- hundreds.) 

Read  the  following  numbers  : 
1.  432.         2.  395.  3.  216.  4    743. 

5.  341.         6.   704.  7.  435.  8    914. 

9.  370.       10.  891.  II.  706.         12.  289. 

13.  514.       14.  780.  15.  981.         16.  709. 

Teach  the  pupils  that  the  fourth  place  rep- 
resents thousands  (or  is  named  thousands.) 
Teach  the  pupils  to  write  a  comma  between 
hundreds  and  thousands  before  reading  a  num- 
ber. 

Read  : 
17.   1395.      18.  3741.      19.  5416.       20.   7308. 
21,  9150.      22.  7075.      21.  4118.      24.  9400. 
25.  7804.      26.   1000.      27.  7050.      28.  8004, 

60.  Teach  pupils  to  write  numbers  to  1,000  in 
Arabic,  and  in  Roman  to  XX. 


40  FIRST  STEPS  AMONG  FIGURES. 

1.  Write  two  hundred  forty-five  (in  Arabic.) 

2.  "  five  hundred  sixty  one. 

3.  "  one  hundred  thirty  four. 

4.  "  seven  hundred  twenty-one. 

5.  "  three  hundred  eighty-six. 

6.  "  four  hundred  sixteen. 

7.  "  nine  hundred  twentj'-one. 

8.  "  six  hundred  thirty-two. 

9.  '*  eight  hundred  seventy-nine. 

61.  Give  more  examples  like  the  above  until 
the  pupils  write  them  readily.  Then  give  the  fol- 
lowing : 

10.  Write  three  hundred  nine. 

11.  "  eight  hundred  forty. 

12.  "  four  hundred  fifteen. 

13.  *'  six  hundred  thirty-seven, 

14.  *'  two  hundred  ninety. 

15.  "  sixty-four. 

16.  "  seven  hundred  two. 

17.  "  five  hundred  six.  .4* 

18.  "  three  hundred  thirty. 

19.  "  four  hundred  one.  'i» 
Review  these  often. 

62. 
3+2  +  3-f3  +  2  +  2+3-f-3-f3  +  3+a  +  a? 
Ans.  31. 


FIRST  STEPS  AMONG  FIGURES.  4I 

4  +  2-1-3  +  3  +  3  +  3+3  +  3  +  2+2  +  2+j 

+  2  ?     Ans.  27. 

a +3 +  3 +  2  +  2  +  3 +3 +3-^3  +  2  +  2? 

Ans.  28. 
4—2x3—4+1x2-3  +  2   I    1H-3  X3  +  2 

+  2+3?     Ans.  13. 
6-7-2-1x3—4+1x2+3  +3+3  +  2  +  2 

+  3+3I     Ans.  25. 
7—4x2  —  2+3+3+34  2  +  2  +  2  +  2+3  + 

3  +  2  ?     Ans.  29. 

5  +  3  +  3+3  +  2+2  +  2  +  2+3+3  +  3  +  2? 

Ans.  ^z- 
7-5x3-2-4-2+3  +  2+  2+3  +  3+3  +  3 
+  2  +  2  ?     Ans.  25. 

Some  boys  are  sliding  down  hill.  There  are 
3  sleds  and  two  boys  on  each  sled,  how  manj' 
boys  are  there  ? 

James  had  7  apples ;  he  ate  one  and  gave 
his  sister  half  of  the  rest.  How  many  did  he 
give  his  sister.^ 

John  had  two  apples  ;  he  cut  each  of  them 
in  halves.     How  many  halves  had  he  ? 

How  many  horses  in  2  two-horse  teams  ? 

A  stingy  boy  had  5  sticks  of  candy  ;  he 
would  neither  eat  any  nor  give  any  away.  How 
many  did  he  keep  ? 


42  FIRST  STEPS  AMONG  FIGURES. 

A  generous  boy  had  3  sticks  of  candy  ;  he 
gave  his  sister  2  sticks  and  he  ate  half  a  stick. 
How  many  had  he  left  ? 

Ralph  had  5  peaches  ;  he  gave  2  of  them  to 
bis  little  sister,  I  to  his  father,  i  to  his  mother^ 
and  ate  I  himself.     How  many  had  he  left  ? 
64.    Count  by  3's  from  3  to  50. 
Count  by  2*s  from  2  to  60. 
Count  by  2's  from  I  to  61. 
Teach  Roman  notation  to  XXVni. 
Write  in  letters  19,  13,  21,  14,  11,  8,  16,  25, 

12,  26,  17,  9,  25,  27,  18. 
66.   Count   by  3*s  from  I  to  22. 
Review  counting. 


EIGHT. 


67.  Pupils  may  make  the  schedule  like  pre- 
vious ones. 

68.  Give  only  a  small  part  of  these  series 
each  day  and  give  with  it  slate  examples  from 
pp.  93  to  94,  and  oral  exercises  in  the  exam- 
ples following  the  series.  By  this  variety  much 
mo.e  work  may  be  accomplished  without  the 
weariness  resulting  from  too  much  sameness. 

The  series  should  on  each  succeeding  day  be 


FIRST  STEPS  AMONG  FIGURES. 


45 


reviewed  ♦    For  a  review  after  completion  take 

the  re-arranged  series. 

For  addition. 

a 

b 

c 

d 

c 

f 

g 

352 

413 

524 

132 

413 

5 

241 

»23 

41  2 

341 

231 

234 

I 

234 

For  subtraction. 

a 

b 

c 

d 

e 

f 

g 

653 

636 

852 

857 

476 

475 

4 

4  I  2 

3  1  2 

3  2  I 

4  3  2 

34  1 

234 

X 

For  multiplication. 

a 

b 

c 

d 

e 

4  2 

I  4 

2  3 

I  3 

2  1 

I  2 

I  2 

3  I 

2  2 

4  3 

For  division. 

a 

b 

c 

d 

43  4 

I  82 

632 

638 

I  3  2 

1  2  I 

3  1  2 

234 

For  addition.     Re-arranged. 

a 

b 

c 

d          e 

f 

g 

142 

3M 

253 

M2     531 

425 

s 

123 

I  23 

41  2 

341     234 

I  23 

4 

For  subtraction. 

a 

b 

c 

d 

e 

f 

g 

345 

627 

665 

484 

875     . 

567 

3 

2>3 

213 

41  2 

342 

341     ' 

432 

1 

*  Leave  on   the    board   the  previous  day's  lesson  in  seriea 
and  add  to  it  as  much  morf  ofthc  series  as  can  be  mastered 

with  the  review. 


44  FIRST  STEPS  AMONG  FIGURES. 


69.   Teach  Roman  notation  to  XXX. 

What  2  equal  numbers  make  8? 

What  is  half  of  8? 

What  tiumber  is  one  less  than  half  of  8  ? 

What  number  can  you  double  and  get  8  ? 

From  what  number  can  you  take  2x3  and 
have  I  left  1 

Count  by  3's  from  i  to  40. 

I  write  a  number  four  limes  JUid  add.  I  get 
8,  what  is  the  number  ? 

Henry  had  half  of  8  cents.  How  manv  had 
be? 

8  cents  will  buy  how  many  l-cent  stamps  ? 

4-cent stamps  (I.  Revenue)?    2-cenl  starapsf 
5cent  stamps  ?     3  cent  stamps  ? 
3  lemons  is  *  of  how  many  lemons  ? 

Lewis  brought  6  eggs  from  the  barn  ;  he  broke 
half  of  them.  How  many  whole  ones  were 
left? 

George  has  3  cents,  he  finds  2  cents  ;  how 
many  must  he  earn  to  have  8  cents? 

William  had  2  sticks  of  candy,  he  ate  half  of 
a  stick,  and  his  sister  half  a  stick  ;  how  much 
candy  had  he  left] 

What  number  is  I  less  than  half  of  6  ? 

Jane's  bird  hatched  3  young  birds  and  there 


FIRST  S>T.  PS  AMONG  FIGURES. 


were  2  eggs  which  did  not  hatch  ;   how  many 
eggs  in  the  nest  at  first  ? 

What  is  half  of  the  number  I  less  than  7  / 
William  bou<jht  3  marbles  at  2   cents  each  ; 
how  much  should  he  pay  for  them? 

How  many  tops  at  3  cents  apiece  can  Ed- 
ward buy  for  8  cents?     For  7  cents? 

6  cents  is  4  cents  more  than  Robert's  money, 
how  much  money  has  he  ? 

3x2—4  +  3  —  2x1  —  1x3^1=?    Ans.  7. 
7— 3  +  2-J-2  — I  X4'-3  is  how  many   less 
than  8?     Ans.  3. 

2  +  3  — 1-^2  — I  x6-i-2-f  5  — 2-1-3  is   *   of 

what  number?     Ans.  4. 

6f2  — 5  +  1-^2-1-1X2    is    I    less   than? 
Ans.  7. 

5  +  2-1-^3x4+2  +  3  +  3  +  3  +  3  +  3  +  2  + 

2  ?     Ans.  23. 

3  +  3-^3-1x3+3  +  3  +  3  +  3  +  3  +  2  +  2  + 

3  ?     Ans.  25. 

7  — 2  — 1-5-2x4  +  2  +  1+3  +  3  +  2  +  2  +  2  + 

2+3  +  3?     Ans.  31. 
5  +  3-^4x3  +  3+  3+3+3  +  2  +  2+3  +  1 

+  3  +  2  ?     Ans.  31. 
8-+2- IX  2-5-3 +  3 +  3 +  3 +  2 +  2 +  3 +  3  + 

3  +  2  +  3  ?     Ans.  29. 


46  FIRST  STEPS  AMONG  FIGURES. 

• 

3  +  2  +  2+3-1-3  +  24-2    +  2    +3   +3    +2? 

Ans.  27. 

4  +  3  +  2  +  2 +3  ■+- 3 +3  +  2  +  2 +3 +3 -I- 2  + 

2  ?     Ans,  34. 

5  +  3  +  2+3  +  3  +  2+3+3  +  2  +  2+2  +3? 

Ans.  33. 
4+2+3+3+3+2+2+3+3+2+2? 

Ans.  29. 
7+3+3  +  3+3  +  2+2+3+3+3  +  2  +  2? 

Ans.  36. 
4+2+3+3+3+2+2+2+2+3+3+3+ 

3  +  3?     Ans.  38. 

5+3-T-2  +  2+3X2+3+3+3  +  3+3-I-2  + 
2  +  2  +  2?     Ans.  27. 
Ora/  exercise.     Place  the  figures  of  any  of 
these  examples  which  involve  only  addition  in 
a  column  on  the  bgard,  let  one  pupil  add  them 
upward,  then  another  downward,  then  another 
add  them  upward,  but  commence  with  I  and  so 
get  a  result  I  greater  than  before,  add  down- 
ward in   the  same  way  commencing  with  one  ; 
then  use  2  instead  of  i  and  also  3  instead  of  2. 
This  makes  8  different  examples  instead  of  one 
and  gives  excellent  practice. 
How  many  feet  have  2  pigs  ? 
3  is  2  more  than  what  number  f 


FIRST  STEPS  AMONG  FIGURES.  47 

8  is  2  more  than  twice  what  number  ? 
How  many  legs  have  3  hens  ? 
What  number  is  5  less  than  8  ? 
How  many  times  can  you  substract  3  from  8  ? 
How  many  legs  have  a  rabbit  and  a  bird  to- 
gether ? 

8  peaches  are  how  many  times  2  peaches  ? 

1  pencil  is  what  part  of  8  pencils  ? 
How  many  wheels  have  4  sulkies  ? 
What  number  must  I  add  to  5  to  get  8  ? 
4  is  half  of  what  number  ? 

From  what  number  can  you  take  2  times  2 
and  have  3  left  ? 

2  is  how  many  less  than  6  ? 
Count  by  2's  from  i  to  62. 

"        "  3's    "     3  to  60. 

«       "  2's     "     2  to  60. 

"       "  3's    «     I  to  61. 
Teach  Roman  notation  to  XXXIX. 
Write  in  letters  26,  18,  34,  9,  16,  22,  37,  19, 

35'  21,36,  39»  17- 

7  is  how  many  more  than  5  ? 

John  had  7  sticks  of  candy;  he  ate  3  and 
gave  away  j4  of  the  rest.  How  many  did  he 
give  away  ?     How  many  did  he  keep  ? 

Count  by  3's  from  2  to  62. 


48 


FIRST  STEPS  AMONG  FIGURES. 


Teach  Roman  notation  to  L. 

Write  in  letters  34,  16,  25,  19,  30,  28,  34, 
17.  26,  ss*  46,  37,  22,  50,  44,  27,  33,  49. 

Write  in  figures  XVIII,  XXXIV,  XV,  XL, 
XXVI,  XII,  XXXVI,  XIV,  XVII,  XIX. 


NINE. 

Pupils  make  a  schedule. 
For  addition, 
a     I      b 

4151324 

1231412 


c 

153 

345 

d 
624 
I  23 

e 
136 
41  2 

f 
241 

345 

h 
451 
511 

532 
235 

J 

g 
362 

234 


For  subtraction. 


a 

b 

c 

d 

e 

f 

736 

498 

756 

948 

585 

384 

412 

345 

125 

312 

341 

232 

h    1   i 

k 

672  1  69 

32 

i|  15 

757 
345 


For  multiplication. 


a 

24 
I  2 


b 

c 

d 

e 

f 

g 

h 

I  0 

2  1 

30 

2  I 

30 

24 

130 

31 

24 

I  2 

31 

23 

42 

234 

FIRST  STEPS  AMONG  FIGURES. 


49 


For  division. 


a 

b 

c 

d 

e 

f 

g 

h 

30 

44 

30 

61 

84 

94 

42 

062 

31 

24 

12 

31 

22 

32 

12 

321 

Re-arranged  and  including  o. 
For  addition. 


a 

b 

c 

d 

e 

i 

■ 

g 

526 

301 ' 

452 

^31 

740 

523 

214 

^23 

435 

^23 

051 

226 

341 

723 

i 

k 

1 

236 

14 

520 

34 

For  subtraction. 


a 

b 

c 

d 

e 

f 

g 

268 

643 

485 

756 

796 

467 

967 

123 

412 

453 

215 

430 

213 

465 

i 

k 

1 

835 

6839 

4 

04 

3 

215  1 

For  multiplication. 


a 

b 

c 

d 

e 

f 

g 

h 

31 

20 

31 

20 

31 

24 

14 

20 

12 

42 

23 

12 

34 

22 

II 

14 

For  division. 

a 

31 
4 


h 

365 
614 


h 

759 
123 


b 

c 

d 

e 

f 

g 

h 

09 

40 

61 

86 

04 

82 

343 

23 

41 

21 

23 

41 

42 

121 

5©  FIRST  STEPS  AMONG  FIGURES. 

How  many  i*s  make  9  ? 

From  what  number  can  I  take  9  I's  and 
have  nothing  left  ? 

How  many  3  cent  stamps  can  you  buy  for  9 
cents  ?  2  cent  stamps  ?  i  cent  ?  5  cent  ? 
4  cent  ? 

How  many  oranges  at  9  cts.  each  can  you 
buy  for  9  cts.  1 

How  many  times  can  I  take  2  from  9  and 
have  I  left? 

George  had  9  peaches  ;  he  ate  one  and  gave 
you  half  of  what  were  left,  how  many  did  he 
give  you  ? 

What  number  taken  3  times  will  make  9  ? 

Henry  gave  each  of  his  3  playmates  3  plums. 
How  many  did  he  give  away  ? 

What  is  half  of  the  number  i  less  than  7  ? 

William  has  9  wheels.  He  has  wheels  for 
how  many  three-wheeled  velocipedes  ? 

5  and  how  many  make  9  ? 

What  number  taken  twice  and  3  added 
makes  9? 

What  is  half  the  number  i  less  than  9  ? 

Theodore  had  9  marbles,  he  lost  4  of  them, 
how  many  had  he  left  1 

4  +  3+ -=9. 


FIRST  STEPS  AMONG  FIGURES.  51 

3  is  3  and  how  many  ? 

1-3  of  6  and  how  niany  make  6  ? 

Count  by  4's  from  4  to  20.     Review. 

AVhat  number  is  i  less  than  ^  of  8  ? 

9=6+  ? 

5  from  9  leaves? 

What  cost  4  lemons  at  2  cts.  each  ? 

What  cost  2  marbles  at  3  cts.  each  ? 

5  +  3—4  —  2  +  3-1-4—6=?     Ans.  3. 

3+4—2+4-^3+3-^24-1=?     Ans.  4. 

6+3  —  54-4—1X2  +  7  +  2=?     Ans.  9. 

2X3+2-5+1^2  +  3-4  +  2+5=  ? 
Ans.  8. 

4-3  +  5-^3+7-^3X2+1-4=?  Ans.  3. 

€+2-6x4-5X3+3  +  3+2+3  ?     Ans. 
20. 

3  +  5-^4+4  4-  2  +  2-f2-f  3-1-3+2-1-2  + 

3+3+2  ?     Ans.  25. 
2X3-3X3-14-2  +  2  +  2+1  +  3+3  +  3 

+  3+2  +  2—4?     Ans.  21. 

5+3+3+2+3  +  2 +2  +  3+3+3  +  2 -»-3 
+  2+3?     Ans.  41. 

^+3  +  3  +  3+2+2+3  +  3+2  +  3+2+2 

+  2+3?     Ans.  39. 
7+2+3+3+2+2+2+3+3+3+2+3 
+  2  +  3  +  2  ?     Ans.  42. 


52  FIRST  STEPS  AMONG  FIGURES. 

3  +  2  +  2+3  +  2+3  +  3  +  2+2+3   +  3-1-2 

+  2  ?     Ans.  32. 

5  +  2  +  5  +  3  +  2+2+3+3  +  3  +  3+2+5 

+  2?     Ans.  38. 
8  +  3  +  2  +  2  +  3  +  2+3  +  3+2  +  2  +  3+3 

3  +  3?     Ans.  42. 

4  +  3  +  2  +  3+3+  3+2  +  2+3+2+3  +  5 

2  +  2  ?     Ans.  37. 

Review  these  often. 

Count  by  2*s,  commencing  with  2  and  with 
I,  to  60  and  61. 

Count  by  3*5,  commencing  with  3,  2  and  i^ 
to  51,  53  and  52. 

Count  by  4's,  commencing  with  4,  to  32  and 
back. 

James  having  9  apples  ate  i,  and  gave  the 
rest  to  his  sisters,  giving  them  2  each  ;  how 
many  sisters  had  he  ? 

George  had  9  oranges ;  he  ate  one  of  them  ; 
if  he  were  to  give  you  half  of  what  were  left, 
how  many  would  you  get  ? 

Charles  having  9  pears  sold  3,  and  gave  his 
sister  half  of  what  he  had  left ;  how  many  did 
he  give  his  sister? 

Teach  Roman  notation  to  LXX. 

Write  in  letters  67,  44,  36,  59,  62,  46,  28,  16. 


FIRST  STEPS  AMONG  FIGURES.  53 

Write  in  figures  XVII,  LXV,  LXVII, 
LXIV,  LXIX,  XXIV,  XIX. 

Pupils  read  the  following,  which  should  be 
copied  on  the  black  board,  and  the  pupil  who 
reads  any  number  to  point  off  the  periods  him- 
self before  reading. 
I.  6305.  2.  7020.  3.  8005.  4.  9400.  5.  1641. 
6.6780.  7.5416.  8.8605.  9.5400.10.7508. 
11.4870.  12.5718.  13.5851.  14.6504.  15.5790- 
19.  1432.  17.9007.  18.5000.  19.7400.  20.8040. 
21.  4637.  22.  5819.  23.  7990.  24.  7803.  25.  7001. 

Teach  Arabic  notation  to  10,000. 

Teach  the  pupils  carefully,  as  being  of  the 
utmost  importance,  that  they  should  place  a 
comma  after  the  number  expressing  thousands 
and  before  they  write  the  units  period.  Teach 
them  that  units  period  takes  three  places,  and 
show  them  that  when  the  number  does  not  fill 
the  three  places,  the  places  on  the  left  must  be 
filled  with  ciphers. 

The  following  are  to  be  read  by  the  teacher 
and  the  pupil  is  to  write  them  in  Arabic.     Re- 
view these  often. 
1-4,573    2.3,240    3.5,296    4-7»3iS 
S.  2,324    6.  7,560    7. 1,427    8.  3,670   9.  7,305 
10.5,741  II.  2.816  12.5,980  13.6,407  u.4,300 


54  FIRST  STEPS  AMONG  FIGURES. 


15.  9,706  16.  4,315  17-  8,590  i8-  i»73i 

19.  7,800  20.  5.004  21.  3.060  22.  5,104 
28.  6,003  24.  8,600  25.  9,419  26.8,040 
27.  5,900  28.  4,307  29.  5,009  30.9,016 
31.  9.360  ^2.  9,070  33.  7,049  34.  3,900 
35.  7,008  36.  7,080  37.  5,700  38.  7,000 

39-  5.875  40.  3,716  41.9,060  32.  5,800 
43.  6,904  44.  6,008  45.  7.600  46.  8,009 

Cut  an  apple  into  3  equal  pieces  and  teach 
the  pupils  that  we  call  one  piece  one  third. 
Break  a  stick  of  candy  into  3  equal  pieces  and 
so  illustrate  the  same  thing. 

In  the  same  way  teach  one-fourth  by  4  divi- 
sions instead  of  3  ;  then  one-fifth,  one-sixth, 
&c.  When  the  pupils  are  familiar  with  this 
show  them  that  of  any  3  equal  things,  one  of 
those  things  is  one-third  ;  of  4  equal  things, 
one  of  them  is  one-fourth.  Do  not  leave  this 
subject  until  the  pupils  are  very  ready  with 
their  answers  to  the  following  questions  : 

An  apple  is  cut  in  5  pieces ;  what  do  we  call 
1  piece  ? 

An  apple  is  cut  in  3  pieces  ;  what  do  we  call 
I  piece  ? 

An  apple  is  cut  in  6  pieces  j  what  do  we  cal) 
I  piece  ? 


FIRST  STEPS  AMONG  FIGURES.  55 

An  apple  is  cut  in  lo  pieces ;  what  do  we  call 
I  piece  ? 

An  apple  is  cut  in  4  pieces  ;  what  do  we  call 
I  piece  ? 

One  apple  is  what  part  of  7  apples? 

One  apple  is  what  part  of  9  apples  ? 

One  apple  is  what  part  of  6  apples  ? 

One  orange  is  what  part  of  4  oranges'? 

One  pencil  is  what  part  of  8  pencils  ? 

A  boy  having  5  apples  gave  away  one  of 
them  ;  what  part  of  his  apples  did  he  give 
away  ? 

A  little  girl  had  6  peaches  :  she  gave  one- 
third  of  them  to  her  brother.  How  many  did 
she  give  him  ? 

(Teach  the  pupils  that  they  get  one-third  of 
a  number  by  dividing  by  3  ;  i  of  a  number  by 
dividing  by  4,  &c.) 

A  boy  gave  away  one-fourth  of  his  4  mar- 
bles ;  how  many  did  he  give  away  }  How  many 
did  he  keep  ? 

If  I  divide  6  apples  equally  among  3  boys, 
what/^r/  of  them  do  I  give  each  boy  ?  How 
maMy  do  I  give  each  boy  ? 


TEN. 

Pupils  make  schedule. 


56 


FIRST  STEPS  AMONG  FIGURES. 


For  addition 

a  I  b 

536247 
2345-2 


c  !  d 

503247 
364623 


e 
535 


f  I  g  I  h 
245623043 


6234 


4S0235I236I74213405 


For  subtraction. 


a    I  b  I  c 
io98|758'676 
623I406I223 
For  multiplication, 
a      b      c  i  d 


6i07;io74  389  1085 
6  45^  5321054    323 


h 
983 
643 


5969 
2345 


f 


352313  240  132  421521 


g 
20 


123  012  415  231  203  124!  53 
For  division. 

a  I  b  I  c  j    d    '  e      f 
842  92oJ6r6  1038J035  44 
4121313  213!   212I231   24 

Re-arranged. 
For  addition. 

a|b|cld|e|f|g|h       i 
635'264  635|207  463  257  430I524  3524 
234  502  345.1652!345!203'464  2352346 
For  subtraction. 

a|b|c|d|e|f|g|h|     i 
8410  8391796  86957101897  75917105  86510 
52  31406543223154  5  623;235!o  42345  6 
For  multiplication. 

i 

31 
31 


a 

b 

c 

d 

e 

f 

g 

h 

42 

14 

24 

3^ 

25 

02 

31 

25 

12 

12 

30 

12 

,41 

35 

23 

12 

FIRST  STEPS  AMONG  FIGURES. 


57 


a 

b 

c 

d 

e 

f 

R 

h 

18 

60 

32 

85 

44 

02 

106 

310 

12 

34 

12 

41 

12 

31 

22 

3  5 

For  division. 

i 

94 
34 

What  is  i  of  10  buttons  ? 

Lucy  had  9  pins ;  she  lost  4  of  them  and 
then  found  i  ;  how  many  had  she  then? 

4  and  how  many  make  10  ? 

10  boys  were  out  in  a  sail  boat  ;  i  more 
than  half  of  them  were  drowned  ;  how  many 
were  drowned  ? 

Minnie  had  9  cents  ;  she  spent  5  of  ihem 
and  lost  4  ;  how  many  had  she  left? 

10  cents  will  buy  how   many  3  cent  stamps? 

How  many  2  cent  sticks  of  candy  can  you 
buy  for  10  cents  1 

How  many  cents  will  4  two-cent  marbles 
cost .' 

Lucy  had  5  cents  and  her  mother  gave  her 
4  cents,  how  many  cents  had  she  then  } 

Show  the  pupils  that  we  call  2  and  2  equal 
numbers,  that  they  are  equal  to  each  other, 
also  I  and  i  are  equal  numbers,  3  and  3,  &c.  ; 
2  and  1,  or  2  and  3,  or  i  and  3  are  unequal 
numbers. 

What  2  equal  numbers  make  4  ? 

What  2  unequal  numbers  make  4  ? 


58  FIRST  STEPS  AMONG  FIGURES. 

What  equal  numbers  make  3? 

What  unequal  numbers  make  3  ? 

What  2  equal  and  i  unequal  numbers  make 
5  ?     (Two  answers.) 

How  many  wheels  have  5  sulkeys  ? 

How  many  wheels  have  3  three-wheeled 
velocipedes  ? 

Mary  was  bringing  in  10  eggs  in  her  apron, 
she  broke  2  less  than  half  of  them  ;  how 
many  did  she  break  ? 

How  many  were  unbroken  ? 

John  had  a  string  10  yards  long  and  William 
had  one  three  yards  long ;  how  much  longer 
was  John's  string  than  William's. 

6  and  how  many  make  10? 

What  number  taken  from  10  leaves  7? 

What  cost  5  two-cent  stamps  ? 

Henry  had  10  miles  to  walk,  he  has  walked 
4  of  them  ;  how  much  farther  has  he  to  walk  ? 

10  beans  are  how  many  times  2  beans  ? 

8  boys  were  playing  "  snap  the  whip,"  6  of 
them  kept  hold  of  hands ;  how  many  were 
there  that  did  not  let  go  ? 

What  number  added  to  3  will  make  10? 

Count  by  4's,  commencing  with  4,  to  60. 


FIRST  STEPS  AMONG  FIGURES.  59 

Review  counting  by  2*s  and  3*5. 
For  rapid  solving. 

4  +  3  — '-^2X3  +  iH-5X4  +  4-^  2+3  +  3 

+  3  +  3  +  2  +  2  +  2=.^     Ans.  24. 
7  +  2  —  3+4^2  —  2x2  +  3  +  3+3  +3  +  2 
+  2+3  +  3  +  4  +  4  +  4=?     Ans.  40. 

4  +  3  +  3  +  3  +  3  +  3  +  2  +  2  +  1+4+4+4 

+  4  +  4=  ?     Ans.  44. 

3  +  1+4  +  4  +  4  +  4  +  4+  3  +  3+3-^2+2 

+  3  +  3  +  2  +  3=  ?     Ans.  48. 

5  +  5-7-2+3-+2-2X3  +  2+  4+4  +  4  +  3, 

+  3+3-^3  +  2  +  2=?     Ans.  36. 
7  +  3  +  3  +  3  +  4  +  4  +  4  +  4  +  3  +2+2+3. 

+  3  +  3  +  3=?     Ans.  51. 
5+3-=-4  +  4-^2+3  +  3  +  3  +  2  +  2  +  3+5 

+  3  +  3  +  2  +  2  +  3  +  3=?     Ans.  38. 

4  +  3+3  +  3  +  3  +  3+  3  +  2+4+4+4  +  4 

+  3  +  2  +  2  +  2+3  +  3=?     Ans.  55. 
10-^5^3  —  2x2-^2  —  2x3  +3  +  2  +  2  + 
3  +  1+4  +  4  +  4=1     Ans.  29. 

5  +  3+4  +  4  +  4  +  4  +  4  +  4  +  4  +  4  +  3+3 

+  3  +  2+2  +  2  +  2  =  ,?     Ans.  57. 

7  +  2  +  2  +  3  +  2+4  +  4  +  443+3+3  +2 
+  2  +  3-1-4  +  4  +  3=?     Ans.  55 

5  +  3 +  4 +  4 -»-4  + 4  + 3  +  2 +3 +4  +  8 +  8 
+  4  +  8  +  2+8=?     Ans.  74. 


6o  FIRST  STEPS  AMONG  FIGURES. 

4+3+2+1  4-4+4+4-1-44-3-1-2+3  4-3 
+  2  +  3+3+3+3=  ?    Ans.  51. 
For  more  practice  see  pp.  51  and  45. 
Count  by  4's  from  2  to  34. 

"       ''   4's     "     4  "  60. 

u       u   2's     "     3  «  60. 

«       "   3's     "      I  "  61. 

u  a     ^'5       u        2    "    62. 


<( 


2  ->> 


60. 


"  *•    2'S      "        I    "    60. 

Write  in  letters  36.  41,  16,  64.  56.  47.  69. 

Write  in  figures  XLV,  XV,  LXIV,  LXXXV, 
XXVII. 

What  equal  numbers  will  make  6?     (3  ans.) 

What  unequal  numbers  will  make  6  ? 
(Several  answers.) 

What  2  equal  numbers  and  i  unequal 
number  will  make  6  .' 

The  following  numbers  are  to  be  copied  on 
the  blackboard  and  the  pupils  are  to  be  re- 
quired to  point  them  off  in  periods  and  read 
them. 


I. 

91017. 

2. 

86700. 

3- 

90007. 

4. 

14071. 

5- 

70000. 

6. 

50010. 

7- 

38419- 

8. 

74058. 

9. 

60800. 

10. 

16040. 

II. 

3000. 

12. 

7014. 

13- 

10061. 

14. 

3020. 

15. 

7003. 

16. 

8500. 

17- 

17500. 

18. 

3540. 

19- 

67374. 

20. 

86000. 

FIRST  STEPS  AMONG  FIGURES.  6f 

What  2  equal  numbers  and  i  unequal  num- 
ber will  make  7  ?     (3  answers.) 

What  3  equal  numbers  and  i  unequal  num- 
ber will  make  7?     (2  answers.) 

What  is  1-3  of  9  ? 

I  is  what  part  of  7  ? 

Teach  Arabic  notation  to  99,000. 

Show  the  pupils  that  they  should  place  a 
comma  after  the  figures  that  express  thousands 
before  writing  the  unit  period,  and  a  period  at 
the  end  of  the  number. 

The   following   numbers  are  to  be  read  by 

the  teacher  and  written   upon  the  blackboard 

or  slates  by  the  pupils. 

I.    7.300.     2.    5,006.     3.    2,050.     4.  10,091. 

5     8,016.     6.    4,000.     7.  12,090.     8.  50,700. 

9.65,078.   10.45,913.   11.80000.   12.80010. 

13.  15,061.   14.40.002.   15.  79,500.   16.81,018. 

17.  30,600.    18.  60060.   19.  90,004.  20.    8,050. 

21.  75,000.  22.  74.695.  23.31,280.  24.  13.300. 

25.  14  041.  26.  10,010. 

Teach  Roman  notation  to  C. 

Write  in  Roman  i.  64.   2.  49.    3.  97.    4.  76. 

Write  in  Arabic  5.  XIX.  6.  LXXXIV.  7. 
XLI.  8.  XXVII.  9.  XVIII.   10.  XXIX. 

Write  in  Roman  11.  17.  12.  56.  13.  83.  14. 
49. 


62  FIRST  STEPS  AMONG  FIGURES. 

Write  in  Arabic,  15.  LXVIII.   16.  XCII. 

If  the  teacher  prefers  it,  the  pupils  can  buy 
their  books  (the  Pupils'  Edition)  at  this  stage, 
and  do  more  slate  work  than  the  teacher  could 
have  time  to  dictate  to  them,  or  copy  upon  the 
board  for  the  pupils  to  copy  and  solve.  If  the 
pupils  do  not  have  their  books  the  teacher  will 
assign  daily  lessons  from  page  i,  2,  &c.,  of  the 
Pupils'  Edition,  doing  it  in  connection  with  this 
work  and  thus  canying  on  that  work  together 
with  the  following  work. 

TAf  teacher  should  how  begin  to  give  the  par- 
allel  work  of  the  Pupils'  Edition  in  connection 
with  that  of  the  Teachers'  Edition.  The  parallel 
pages  are  denoted  by  the  numbers  at  the  bottom 
9/  the  pages  in  each. 


FIRST  STEPS  AMONG  FIGURES. 


63 


PART  II, 


Counting  two  or  more  numbers  into  one 
number  is  called  Adding,  or  Addition. 

The  number  obtained  by  counting  two  or 
more  numbers  into  one  number  is  called  the 
sum  of  those  numbers. 

For  addition  and  multiplication.  (5  and  rev.) 


a 

b 

c 

d 

e 

f 

g 

352 

413 

524 

135 

241 

352 

4  I 

234 

523 

452 

345 

234 

523 

4  5 

For  subtraction. 

a 

b 

c 

d 

e 

f 

g 

7410 

746 

758 

693 

696 

857 

85 

32  5 

432 

523 

452 

345 

432 

5  4 

For  division. 

a 

b 

c 

d 

e 

41510 

6165 

20  92 

20815 

610  8 

4  5  2 

3  45 

4  32 

54  3 

2  5  « 

f 

g 

31225 

4  12 

3  4 

5 

2 

3 

See  Pupils'  Edition,  p.  5. 


64  FIRST  STEPS  AMONG  FIGURES. 

It  is  well  to  require  pupils  to  bring  a  written 
analysis  of  an  example  to  recitation  and  to 
give  the  solutions  of  other  examples  orally  in 
class  in  the  same  form,  but  there  should  be  a 
large  number  of  examples  given,  of  which  only 
the  answer  is  to  be  given  and  that  as  soon  as 
possible  after  the  reading. 

1.  Susan  had  4  cents  and  her  mother  gave 
her  3  more  ;  how  many  had  she  then  ? 

Solution.  She  had  the  sum  of  4  cents  and 
3  cents,  or  7  cents. 

2.  John  has  5  marbles  and  James  has  4  mar- 
bles ;  how  many  have  both  ? 

3.  Lulu  has  3  eggs  in  one  hand  and  2  in  the 
other  ;  how  many  has  she  in  both  ? 

4.  Walter  bought  some  candy  for  4  cents 
and  some  raisins  for  5  cents ;  how  many  cents 
did    he  spend  ? 

5.  Martha  read  4  pages  in  the  forenoon  and 
2  in  the  afternoon  ;  how  many  did  she  read  that 
day  t 

6.  A  boy  had  3  pencils  in  one  pocket  and  5 
in  another  ;  how  many  had  he  in  both  ? 

7.  If  a  top  cost  4  cents  and  a  marble  cost  2 
cents,  how  many  cents  must  a  boy  have  to  buy 
a  top  and  a  marble  ? 


FIRST  STEPS  AMONG  FIGURES.  65 


8.  Jane  bought  2  books  ;  she  had  3  before. 
How  many  has  she  now  ? 

9.  Henry  walked  4  miles  before  dinner,  and 
4  after  dinner  ;  how  far  did  he  walk  that  day  ? 

10.  There  are  3  barrels  of  apples  under  one 
tree,  and  two  under  another ;  how  many  under 
both? 

Ask  the  pupils  to  bring  examples  of  their 
own  to  recitation  different  from  those  given 
them.  The  teacher  also  will  make  additional 
examples,  using  pleasant  facts  about  the  school 
room,  or  the  pupils,  or  iheir  homes,  something 
they  have  seen. 

Count  by  4's  from  2  to  62. 

Taking  one  number  from  another  number  is 
called  subtracting,. or  Subtraction. 

The  number  obtained  by  taking  one  number 
from  another  number  is  called  the  Remainder 
or  Difference. 

1.  Joseph  had  8  cents;  he  spent  5  cents  for 
an  orange.     How  niany  cents  had  he  left  ? 

Solution :  He  had  left  the  difference  be- 
tween 8  cents  and  5  cents,  or  3  cents. 

2.  Mary  had  a  cake  which  she  cut  into  10 
pieces;  7  were  eaten.     How  many  were  left? 

3.  My  knife  has  6  blades;  2  of  them  are 
open.     How  many  are  closed? 

Sec  P.  i:«l.,  p.  9. 


06  FIRST  STEPS  AMONG  FIGURES. 

4.  James  bought  a  paper  for  5  cents  ;  he  gave 
the  newsboy  10  cents.  How  much  change 
should  James  receive  ? 

5.  Samuel  put  9  peaches  on  the  table  and 
his  sister  took  5  of  them  ;  how  many  were  left  ? 

6.  A  man  owed  $7  ;  he  paid  $3  ;  how  many 
|*s  did  he  then  owe  ? 

7.  Willis  took  9  cents  to  buy  candy  with  ;  he 
lost  4  cents.  How  many  had  he  to  buy  candy 
wiih  ? 

8.  Henry  bought  a  pencil  for  4  cents  and 
sold  it  for  7  cents  ;  how  many  cents  did  he 
gain  ? 

8.  Matthew  bought  one  pencil  for  4  cents 
and  another  for  5  cents ;  what  did  both  cost 
him  } 

10.  Susan  bought  3  spools  of  white  thread 
and  6  spools  of  blue  thread  ;  how  many  spools 

^  did  she  buy  ? 

11.  Jane  tried  to  solve  6  examples;  she  had 
4  of  them  correct.    How  many  were  wrong  ? 

12.  Waller  had  8  pencils;  he  broke  3  of 
them.    How  many  whole  ones  had  he? 

13.  Fanny  had  6  needles ;  she  found  4  more. 
How  many  had  she  then  ? 

14.  May  is   7    years   old   and   her   brother 


FIRST  STEPS  AMONG  FIGURES.  67 

Frank  is  4  years  old  ;  how  much  older  is  May 
than  Frank  ? 

15.  George  had  a  stick  9  inches  long  ;  he  cut 
off  3  inches  of  it.    How  long  was  the  stick  then  ? 

16.  A  farmer  having  8  turkeys,  sold  4  o^ 
them  ;  how  many  had  he  left  ? 

17.  John  paid  3  cents  for  candy  and  5  cents 
for  marbles  ;  how  many  cents  did  he  spend  ? 

18.  A  little  boy  had  3  fingers  cut  off  in' a 
machine  ;  how  many  had  he  left  ? 

19.  Silas  had  3  marbles  in  one  pocket  and  5 
in  the  other  ;  how  many  had  he  in  both  ? 

20.  How  many  wheels  have  a  sulky  and  a 
wagon  together  ? 

21.  There  are  3  girls  on  the  front  seat  of  a 
carriage,  and  5  girls  on  the  back  seat ;  how 
many  girls  in  the  carriage? 

22.  Jesse  had  8  sticks  of  wood  to  bring  in  ; 
he  has  brought  in  3  ;  how  many  more  has  he  to 
bring  in  ? 

23.  An  orange  cost  6  cents,  and  a  peach  cost 
3  cents ;  how  much  more  did  the  orange  cost 
than  the  peach  ? 

24.  If  a  pear  cost  4  cents,  and  a  lemon  cost 
5  cents,  what  will  a  pear  and  a  lemon  cost  ? 

25.  There  were  6  eggs  in  a  nest  and  4  of 

See  V.  Ld.,  p.  11. 


68  FIRST  STEPS  AMONG  FIGURES. 

them  were  broken  ;  how  many  whole  ones  were 
there  ? 

26.  Ellen's  father  gave  her  9  cents  ;  she 
bought  a  doll  with  5  cents.  How  many  cents 
had  she  left  ? 

27.  There  are  4  boys  riding  in  a  sleigh  and 
2  riding  behind  on  the  runners  ;  how  many 
boys  with  the  sleigh  ? 

28.  Ella  has  5  roses  on  her  bush,  and  5  in 
her  hand  ;  how  many  has  she  } 

29.  There  were  9  chickens  in  a  coop  and  a 
rat  ate  3  of  them  ;  how  many  were  left  ? 

30.  There  are  in  the  class  4  girls  and  3  boys  ; 
how  many  pupils  in  the  class  ? 

31.  A  little  boy  bought  10  slicks  of  candy  , 
he  ate  3  of  them  and  gave  away  the  rest.  How 
many  did  he  give  away  1 

These  31  examples  should  be  reviewed  and 
others  given,  until  the  pupils  know  at  once  in 
such  simple  problems  whether  they  are  to  find 
the  sum  or  the  difference.  Review  the  series 
also. 

Count  by  4's  from  i  to  17. 

Copy  the  following  examples  one  at  a  time 
on  the  blackboard  ;  require  a  pupil  to  point  one 
off  into  oeriods  and  read   it.     Erase  it,  then 


FIRST  STEPS  AMONG  FIGURES. 


69 


write  another  and  require  another  pupil  to  point 

it  off  and  read 

and  so  with  the  others  : 

I.  90016. 

2-  45378. 

3    461340. 

4.  908714. 

5.  876341. 

6.  608790. 

7.  379000. 

8.  75608. 

9.   40713 

10.   100740. 

II.  98716. 

12.  900000. 

13.  800601. 

14.  200003. 

15.   761300. 

16.  500000. 

17.   700300. 

18.  60050.     , 

19.  700060. 

20.  600000. 

21.   200361. 

22.  500700. 

23.  40010. 

24    900007. 

Teach  Arabic  notation  to  999 

,000. 

To  be  read  by  the  teacher  for  pupil  to  write 

upon  slates  or 

blackboard. 

Write  in  Arabic  the  following 

: 

I.   1,040. 

2.  3,506. 

3.    10,016. 

4.  8,400. 

5.  9.350. 

6.  7,518. 

7.  3,761. 

8.   10,010. 

9.  40.070. 

10.  73,801. 

II.  36,000. 

12.  90,090. 

13.   100,100. 

14.  702,940. 

15.  900,070. 

16.  816,902. 

17.  49»049- 

18.  860,705. 

19.    461,017.     20.    791,486.21.     21.    10020. 

Write  these,  or  similar  numbers  on  the  board 
and  require  the  pupils  to  read  them. 
Write  in  Roman  the  following : 
1.  79.  2.  96.  3.  no. 

4.  47.  5-  '9-  6.  134. 

See  P.  Ed.,  p.  26. 


yo  FIRST  STEPS  AMONG  FIGURES. 


Write  in  Arabic : 

7.  CLx.        8.  xcii.  9.  cxa 

lo.  CXLIV.      II.  CLXXXVI.  12.  CIX. 

Write  in  Roman  : 

13.  192.      14.   136.       15.   168.      16.  154. 

A  short  method  of  adding  equal  numbers  is 
called  Multiplication; 

or, 
Taking  a  number  a  certain  number  of  times  is 
called  Multiplication. 

The   number  obtained  by  multiplication  is 

called  the  Product. 

1.  John  bought  5  pencils  at  4  cents  each  ; 
what  did  they  cost  ? 

Solution  :  They  cost  5  times  4  cents,  or  20 
cents  ;  or  if  one  pencil  cost  4  cents,_;fz^r  pencils 
will  cost  5  times  4  cents,  or  20  cents. 

2.  If  I  orange  cost  5  cents,  what  will  4 
oranges  cost  ? 

3.  What  cost  4  marbles  at  3  cents  each  ? 

4.  How  many  quarts  in  2  gallons  ? 
Solution  :  In  one  gallon  there  are  4  quarts,  in 

2  gallons  there  are  2  times  4  quarts,  or  8  quarts^ 

5.  How  many  pints  in  3  quarts/ 

6.  How  many  quarts  in  5  gallons  ? 

7.  How  many  wheels  have  3  wagons  \ 


FIRST  STEPS  AMONG  FIGURES.  Jl 

8.  A  lady  gave  3  little  girls  5  bunches  of 
grapes  each ;  how  many  bunches  did  she  give 
them  ail  ? 

9.  How  many  feet  have  4  hens. 

10.  How  many  feet  have  3  cows  ? 

11.  What  cost  5  books  at  4  shillings  each? 

12.  What  cost  3  lead  pencils  at  5  cents  each  ? 

13.  If  a  lead  pencil  cost  6  cents  and  a  mar- 
ble cost  3  cents,  what  will  both  cost  ? 

14.  What  cost  a  doll  worth  4  cents,  and  a 
spool  of  thread  worth  6  cents  ? 

15.  What  cost  3  pencils  at  4  cents  each  ? 

16.  The  boys  are  riding  down  hill  on  sleds  ; 
there  are  4  sleds  and  2  boys  on  each  sled  ; 
how  many  are  riding  down  hill  ? 

17.  A  boy  bought  a  sled  for  8  shillings  and 
sold  it  for  5  shillings  ;  how  many  shillings  did 
he  lose  ] 

18.  A  boy  had  8  cents  in  his  pocket,  but  he 
lost  4  of  them  through  a  hole  in  his  pocket ; 
how  many  had  he  left  ? 

19.  A  boy  paid  4  cents  for  candy  and  5  cents 
for  nuts;  how  much  money  did  he  spend? 

Review  these  examples  carefully. 
Count  by  4's  from  i  to  29. 
For  rapid  solving. 

See  P.  Ed.,  p.  19. 


72  FIRST  STEPS  AMONG   FFGURES. 

i2-r-4  +  3+3+3-f-4  +  4-l-4-+-4  +  3-+"3  +  3 

+  2  4-2=  ?     Ans.  41. 
3-1-4  +  4  +  3+3  +  3  +3 -'-2  +  2+2+3+3 

+  2  +3+4=  ?     Ans.  44. 
3  +  3  +  2  +  2  +  2  +  3+3+3  +2+3  +  2  +  2 

+  3+3  +  3=?     Ans.  39 
3+4  +  4  +  4  +  31-3   -   2  +2+4  +  ^  +  3  +  2 

f  2   r  2—  ?       Ans.  41. 
3+3+3+3  +  2+2  +  2  +  2  +3+3  +  2+3 

+  3+4=?     Ans.  37. 
3  +  2  +  2  +  2+3+3+  2+  2  +1+3 -f- 2  +  3 
+  3  +  1=?     Ans.  32. 

4+3  +  4+4—  3+4-3-2  +  4  +  4-3+2 
+  3=  ?     Ans.  21. 

a +  3 +  4  +  4 +  4-3 +  2 -3  -  2-3  +  4  +  2 
+  3=  ?     Ans.  17. 

a+4  +  3+4  +  3  -4-3  -4  +  3+3+4  +  4 

+  3=  ?     Ans.  22. 
5  +  3-2-4  +  3+3  +  2+4  +  3+4  +  3-2 

—3=  ?     Ans.  19. 
3+4  +  2+4  +  2  +  2—4—4+3  —  2—4  +  3 

+  4=?     Ans.   13. 
3  +  2+4-3-2+3  +3+3+3  +  4-3-2 

—4  +  3  +  2=1     Ans.  16. 
4+3-«-2+3  +  3+4+3  -4-4  +  1-3  +  4 

—3-4-4=?     Ans.  5. 


FIRST  STEPS  AMONG  FIGURES.  73 

20.  What  cost  5  marbles  at  4  cents  each  1 

21.  What  cost  a  marble  at  3  cents  and  a 
pencil  at  4  cents? 

22.  James  bought  3  books  at  4  shillings 
each  ;  what  did  they  cost  him  ? 

23.  David  had  8  apples  when  he  started  for 
school,  but  he  ate  3  on  the  way  ;  how  many 
had  he  when  he  got  to  school  ? 

24.  Sarah  ate  3  crackers  at  breakfast  and  5 
at  dinner;  "how  many  did  she  eat  at  both 
meals? 

25.  How  many  horses  in  3  four-horse  teams? 
B'inding    how    many    limes    one   number    is 

contained  in  another  is  called  Divis'on. 

The  number  which  shows  how  many  times  it 
is  contained  is  called  a  Quotient. 

1.  How  many  pears  at  2  cents  each  can  be 
bought  for  8  cents  ? 

Solution :  If  i  pear  cost  2  cents,  for  8 
cents  you  can  buy  as  many  pears  as  2  is  con- 
tained times  in  8,  or  4 ;  or,  as  many  as  there 
are  2's  in  8,  or  4. 

2.  John  has  15  cents  ;  how  many  marbles  at 
3  cents  each  can  he  buy  ? 

3.  Willis  spent  20  cents  for  oranges  at  5 
cents  each ;  how  many  oranges  did  he  buy  ? 

See  P.  Ed.,  p.  25. 


74  FIRST  STEPS  AMONG  FIGURES. 

4.  If  one  pig  cost  $4,  how  many  pigs  may 
be  bought  for  $12  ? 

5.  How  many  lead  pencils  at  4  cents  each 
can  be  bought  for  20  cents  ? 

6.  If  I  doll  cost  3  shillings,  how  many 
such  dolls  can  be  bought  for  12  shillings  ? 

7.  10  shillings  will  buy  how  many  knives 
at  2  shillings  each  ? 

8.  How  many  balls  at  3  shillings  each  may 
be  bought  for  6  shillings  ? 

9.  When  pears  are  2  cents  each,  how  many 
can  you  buy  for  8  cents  ? 

Count  by  4's  from  i  to  61. 
«       u    ^^'s    «t     2  "  62. 

"       "    4's     "     4  "  60. 
«       «    4's     "     3  "  63. 

10.  \Vhat  cost  4  pineapples  at  2  shillings 
each? 

11.  How  many  pencils  at  4  cents  each  can 
be  bought  for  16  cents  ? 

12.  A  boy  walked  5  miles  i  day,  and  3  miles 
the  next  day;  how  far  did  he  walk  in  the  2 
days  ? 

13.  How  many  pairs  of  mittens  at  3  shillings 
a  pair  can  you  buy  for  9  shillings  ? 

14.  20  shillings  will  buy  how  many  purses 
at  5  shillings  each  ? 


FIRST  STEPS  AMONG  FIGURES. 


75 


15.  Henry  earned  9  cents  on  Monday  and  ^ 
cents  on  Tuesday  ;  how  many  cents  more  did 
he  earn  on  Monday  than  on  Tuesday  ? 

16.  Louisa  had  5  cents  and  she  found  5 
cents  more  ;  how  many  cents  had  she  then  ? 

17.  If  one  sled  cost  5  shillings  how  many 
sleds  can  you  buy  for  20  shillings? 

1 8.  What  cost  4  vests  at  $8  each  ? 

19.  How  many  neckties  at  3  shillings  each 
can  you  buy  for  12  shillings  ? 

20.  5  little  boys  each  have  a  pair  of  copper- 
toed  boots  ;  how  many  boots  have  they  ? 

Review  these  examples. 

For  addition  and  multiplication.    (6  and  rev.) 


a 

1 

0 

( 

: 

c 

I 

e 

3624 
2345 

1362 
6345 

4154 
6234 

6241 
5623 

5536 
2456 

f 
2415 
2345 

g 
362 

623 

h 
3  1  5 

45  6 

For  subtraction. 

a 

b              c 

d 

9  7  6  10 
5634 

7  II  7  8     3  10  8  6 
5    543     2    661 

4798 
3245 

e 

12  4 : 

6  22 

1 1 

;  6 

68 

5  4 

f 

58 
3  2 

BCP. 

9 

6 

Ed 

1 

I 

>] 

05 
54 

3.  28, 

5 
2 

h 
9 
3 

6 
4 

76 


FIRST  STEPS  AMONG   FIGURES. 


For  division. 

a 

b 

c 

30 

12  8  3 

10  20  1 

s  36 

12   15  2 

24 

5 

6  2  3 

2     4 

5     6 

4     3  2 

6 

d 

e 

f 

4 

12  4  25 

18   12 

6   16 

5  30  6 

18 

2 

3  4     5 

6     2 

3     4 

5     6  2 

3 

g 

h 

8  20  6 

9  24 

10 

456 

3     4 

S 

<0 

(2)         (3) 

(4) 

(5) 

(6) 

32 

33             2 

23 

33 

31 

13 

21           32 

33 

31 

23 

3 

2           23 

31 

23 

33 

21 

30           33 

12 

13 

23 

13 

13           21 

22 

22 

32 

33 

32           23 

31 

32 

»3. 

"S 

131         134 

152 

154 

155 

{7) 

(8)          (9) 

(10) 

(II) 

(12) 

30 

23           32 

3 

31 

23 

23 

33           30 

23 

33 

32 

33 

30           23 

31 

23 

13 

12 

22             3 

23 

30 

33 

31 

31           31 

33 

22 

21 

23 

33           23 

32 

31 

32 

33 

22           33 

21 

33 

23 

'85 

194         17s 

166 

203 

177 

See  P.  Ed 

.,  p.  16. 

FIRST  STEPS  AMONG  FIGURES.  77 

13.  Add  32,  21,  3,  33,  23,  31,  23,  33. 

14.  Add  S3^  23,  30,  21,  2,  S2,  23,  22. 

15.  Add  2,  33,  22,  31,  23,  32,  12,  23. 

16.  Add  31,  30,  23,  s:^,  21,  32,  22,  32. 

17.  Add  23,  13,  32,  23,  3,  13,  32,  33. 
Count  by  5's  from  5  to  60. 

For  rapid  solving. 

3-f4-f4  +  3  +  3  +  2+4  +  3+3  +  2+4+4 
+  3  ?    Ans.  42. 

2+3+4+3+3+5+5+3+3+4+4+4+ 

3  +  3  +  2  ?     Ans.  51. 

3+4+4+3+3+4+4+5+5+5+3+2+ 

4  +  4  +  4?     Ans.  57. 

3  +  4  +  3+3+3  +  2+2  +  3  +4  +  2  +  4  +  3 

+  2  +  3  +  5?     Ans.  46. 

5+*3  +  4  +  2  +  3+4  +  3  +  3+  2  +4+  3  +4 
+  3  +  3+4?   Ans.  50. 

4  +  3  +  4  +  3-2-2  +  5-3+  2  +4?     Ans. 

18. 
2 +4-+3X  5-4x4 -3-3-^3 +6-^4x6 

+  4  +  3  —  2?     Ans.  23. 
3  +  6  +  4  +  4  +  4  +  3+4-6-4-2-5  +  3 

+  4—4.^     Ans.  14. 
3  +  5  +  2X5-3  +  4-2-4H-5  X6  +  4  +  3 

■+5  +  3  +  3^     Ans.  11. 


78  FIRST  STEPS  AMONG  FIGURES. 

2  +  5  +  4+1-^4+1  X5-4-I- 3  4  4+1-^4 

+  5  +  2+4  +  3?     Ans.  20. 

3  +  6  +  4-1  -3x6-3-4-3  +4^6x5 

+  4  +  3+4  +  3?     Ans.  29. 

5  +  6—3+4  +  3+3  -^3  +  4  +  3+4  +  4-2 
—  3+4  +  3  +  4?  Ans.  27. 
Do  not  use  all  of  these  at  once,  but  use  them 
occasionally  and  in  connection  with  a  lesson 
in  examples  of  another  kind,  or  to  wake  up 
the  whole  school  sometimes  when  they  are 
listless. 

1.  What  cost  a  pencil  at  6  cents  and  a  mar- 
ble at  4  cents  / 

2.  Mary  had  5  peaches  and  her  brother  gave 
her  6  more  ;    how  many  had  she  then  ? 

3.  What  cost  5  books  at  6  shillings  each  ? 

4.  3  boats  are  on  the  lake  ;   each  has  a  pair 
of  oars,  how  many  oars  have  the  3  boats  ? 

5.  There  are  9  boys  in  a  class,  and  6  of  them 
recite  well  ;   how  many  do  not  recite  well  ? 

6.  How  many  baskets,  at  4  cents  each,  can 
be  bought  for  24  cents  ? 

7.  Charles  spent  18  cents  for  candy  at  three 
cents  an  ounce  ;  how  many  ounces  did  he  buy  ? 

8.  Jane  had  10  needles,  she  lost  7  of  them ; 
how  many  had  she  then  ? 


FIRST  STEPS  AMONG  FIGURES.  79 

9.  A  lazy  boy  brought  his  mother  3  sticks  of 
wood  at  one  time  and  4  at  another ;  how  many 
sticks  did  he  bring  her  ? 

10.  Six  boys  can  sit  on  this  seat,  how  many 
boys  can  sit  on  four  such  seats  ? 

1 1.  In  a  school  room  there  are  6  keys  hang- 
ing on  a  nail,  2  keys  for  each  door  ;  how  many 
doors  are  there  ? 

12.  There  are  i8  words  in  the  sp-lling  les- 
son, 3  words  for  each  pupil  ;  how  many  pupils 
in  the  class  ? 

13  There  are  5  piles  of  books  and  6  books 
in  each  pile  ;  how  many  books   in  the  5  piles? 

14.  One  stormy  day  George  cleared  the  path 
of  snow  4  times  in  the  forenoon  and  5  times  in 
the  afternoon  ;  how  many  times  did  he  clear 
the  path? 

15.  Amelia  had  11  cents  and  spent  5  of 
them  ;  how  many  had  she  left  ? 

16.  Arthur  had  7  buttons  on  his  jacket ;  how 
many  had  he  after  losing  2  of  them  ? 

17.  How  many  quarts  in  6  gallons? 

18.  Mr.  Smith  has  a  quart  of  maple  syrup; 
how  many  times  can  he  fill  a  pint  cup  with  it  ? 

19.  How  many  skates  at  6  shillings  each  can 
you  buy  for  24  shillings? 


8o  FIRST  STEPS  AMONG  FIGURES. 

20.  What  cost  a  knife  at  5  shillings  and  a 
saw  at  6  shillings  ? 

21.  If  2  oranges  cost  12  cents,  what  will  i 
orange  cost  ? 

Solution  :  If  2  oranges  cost  12  cents,  i 
orange  will  cost  i  of  12  cents,  or  6  cents;  or 
I  orange  will  cost  i  of  12  cents,  or  6  cents. 

Before  giving  examples  like  the  above  teach 
the  pupils  carefully  that  if  2  things  of  equal 
value  cost  a  certain  sum,  i  of  them  will  cost  J 
of  that  .sum  ;  if  3  cost  that  sum,  i  of  them  will 
cost  1-3  of  it;  if  5  cost  any  sum,  i  of  them 
will  cost  1-5  of  it ;  if  9  of  them  cost  any  sum, 

1  of  them  will  cost  1-9  of  it,  &c. 

Question  on  this  subject  until  it  is  thorough- 
ly mastered. 

Show  the  pupils  that  to  get  J  of  12  apples 
(or  marbles  or  pencils)  they  may  be  placed  in 

2  equal  piles,  and  they  will  find  that  ^  of  12 
apples  is  6  apples. 

Show  them  that  to  get  i  of  12  pencils,  they 
may  be  placed  in  3  piles,  and  that  i  of  12  is  4. 
Show  in  the  same  way  that  i  of  12  is  3.  When 
this  is  thoroughly  understood,  show  them  that  i 
of  12  may  be  obtained  by  dividing  12  by  2  — 
the  result  in  each  case  being  6 ;  show  that  i 


FIRST  STEPS  AMONG  FIGURES.  8 1 

of  12  may  be  obtained  by  dividing  12  by  3  ; 
teach  i  of  12  in  like  manner.  Illustrate  also 
by  J  of  6  and  i  of  6  and  i  of  8. 

Then  teach  in  general  terms  that  i  of  any 
number  may  be  obtained  by  dividing  the  num- 
ber by  2  ;  i,  by  dividing  by  3 ;  },  by  7,  &c. 

22.  What  cost  I  pear  if  4  pears  cost  8  cents? 

23.  If  3  knives  cost  15  shillings  what  will  i 
knife  cost  ? 

24.  If  2  pencils  cost  16  cents,  what  will  i 
pencil  cost  ? 

25.  If  4  stools  have  12  legs,  how  many  legs 
will  I  stool  have  ? 

26.  If  6  boys  earn  18  cents,  how  many  does 
I  boy  earn  ? 

27.  If  5  cords  of  wood  cost  $25,  what  will  i 
cord  cost  ? 

28.  How  many  pounds  in  i  box  of  honey  if 
4  boxes  contain  24  pounds  ? 

29.  At  5  cents  each,  how  many  oranges  can 
be  bought  for  30  cents  ? 

30.  If  3  lemons  cost  18  cents,  what  costs  i 
lemon  ? 

31.  How  many  pounds  of  butter  will  last  a 
family  i  week  if  they  use  12  pounds  in  4  weeks  1 

32.  How  many  days  will  18  apples  last  a 
boy  who  eats  3  apples  each  day  ?  6 


82  FIRST  STEPS  AMONG  FIGURES. 

33.  A  blacksmith  shod  5  horses  each  day ; 
how  many  did  he  shoe  in  6  days  1 

34.  6  boys  are  skating  on  the  ice,  and  4 
lx>ys  are  sliding  on  the  ice  without  skates ;  how 
many  boys  on  the  ice  ? 

35.  Nine  boys  were  riding  down  hill  on  sleds ; 
3  of  them  went  home.  How  many  continued 
to  ride  down  hill  .> 

36.  If  15  yards  of  cloth  will  make  5  pairs 
of  trowsers,  how  many  yards  will  it  take  to 
make  i  pair  of  trowsers  ? 

37.  George  has  4  books,  and  Mary  has  5 
books  ;  how  many  have  both  ?  » 

38.  12  cents  will  buy  how  many  marbles  at 
3  cents  each  ? 

39.  If  5  marbles  cost  10  cents,  what  will  i 
marble  cost  ? 

40.  What  cost  I  apple  if  5  apples  cost  10 
cents  ? 

41.  How  many  dolls  at  4  shillings  can  you 
buy  for  20  shillings  ? 

Review  the  last  21  examples  until  the  pupils 
solve  them  readily  and  can  distinguish  when 
they  divide  and  when  get  one-half  or  one-third, 
&c. 

Count  by  5's  from  i  to  61. 


FIRST  STEPS  AMONG  FIGURES.  83 

Count  by  4's  from  i  to  6i. 
Count  by  4's  from  2  to  62. 
Count  by  3's  from  3  to  60. 
Count  by  3's  from  i  to  61. 
Count  by  3's  from  2  to  62. 
To  be  read  by  the  teacher. 
Write  in  Arabic : 


I. 

7,050.       2. 

10,003. 

3- 

40,300. 

4. 

5.209.       5- 

10,010. 

6. 

300,040. 

7- 

9,610.        8. 

4,316. 

9- 

215,000. 

10. 

80,090.    II. 

600,000. 

12. 

809,740. 

13- 

100,010. 14. 

916,008. 

15. 

835.941. 

16. 

70,000.    17. 

90,005. 

18. 

5,016. 

19. 

213,033.  20. 

30,000. 

Write  these  or  similar  numbers  on  the  black- 
board and  require  pupils  to  read  them. 
For  rapid  solving. 

6  +  4  +  3  +  24-4  +  4-3-^4x6—3-3  +  4 
+  4?     Ans.  32. 

3  +  5  +  4  +  3  +  3+4  +  3+-5X4-4  +  3+3 

-4  +  3  +  5+5?  Ans.  31. 
4+3+4+4+4+2+3+4+4+2+5+2 

+  3  +  3+4-^  Ans.  53. 
3  +  2+4  +  3+44-3  +  2+3+3+4  +  3  +  2 

+  3+3  +  2  1  Ans.  44. 
2+3+3+2+4+4+3+2+3+3+4+4 

+  2  +  3  +  3+4  +  3^  Ans.  52. 


84  FIRST  STEPS  AMONG  FIGURES. 

3+2+4+3+5+3+2+4+4+1+2+4 
+  I+3  +  I  +  2?  Ans.  44. 

4+3+3+4+2+3+2+3+3+3+2+4 
+  I  +  2  ?  Ans.  39. 

3+2+4+4+3+3+2+2+3+4+2+3 
+  3  +  2  +  1?  Ans.  41. 

17  +  4  +  3-^6  +  6  +  5-3  +  3x6  +  3-4  — 

3  +  5x3*     Ans.  12. 
9  +  4  +  3  +  2^3><5-4— 3-2  +  4— 1-^6 

X3+3  +  2  ?     Ans.  17. 
16  +  4  +  3  +  3+3  +  2  — 1-^5 +3+3^3  X 
5-3+4-1-^4?     Ans.  5. 
In  giving  the  following  examples  as  well  as 
those  "  for  rapid  solving  "  the  teacher  should  be 
very  careful    that  pupils    do    not    acquire    a 
pernicious  habit  of  counting  instead  of  adding 
at  sight  or  as  soon  as  heard. 

Slate  examples. 
(,.)     (2)     (3)     (4)     (5)     (6)     (7)     (8)    (9) 

2I22I2I2I 

1  2      I22I2      12 

2  2      II222      22 
2      I      22122      22 

1  2      12222      22 

2  2  22I2I  21 
I              2              22222  22 

I  12222  12 

II  I  I  II  21 


13  12 


16  14         16         15  16         15 


FIRST  STEPS  AMONG  FIGURES. 


8j 


(io)(.. 

)(") 

(I3)(«4)(i5)(i6)(.7)(i8)(i9) 

2 

I 

2 

2 

2 

2 

I 

I 

2 

I 

I 

2 

I 

I 

2 

I 

2 

2 

2 

2 

2 

2 

2 

2 

2 

I 

2 

2 

I 

2 

2 

2 

2 

2 

I 

2 

I 

2 

2 

I 

2 

2 

I 

I 

I 

2 

2 

I 

2 

I 

2 

2 

2 

2 

I 

I 

2 

2 

2 

I 

2 

2 

I 

T 

I 

I 

2 

2 

2 

I 

I 

2 

2 

I 

2 

I 

I 

1 

2 

2 

2 

2 

2 

2 

I 

2 

2 

2 

2 

2 

I 

I 

2 

2 

3 

2 

2 

2 

2 

I 

I 

I 

I 

2 

a 

2 

2 

I 

2 

2 

I 

2 

I 

I 

I 

I 

I 

I 

2 

2 

I 

I 

i8 

i8 

21 

22 

22 

21 

24 

20 

19 

19 

(20) 

[21) 

(22 

) 

(23) 

(24) 

(25) 

2 

I 

I 

2 

2 

I 

2 

2 

I 

I 

3 

2 

2 

2 

2 

2 

2 

I 

2 

2 

2 

I 

2 

2 

I 

I 

I 

I 

I 

I 

I 

2 

2 

I 

2 

2 

I 

2 

I 

2 

2 

I 

2 

2 

2 

2 

2 

2 

2 

I 

a 

2 

I 

I 

I 

2 

I 

2 

I 

I 

I 

2 



I 

I 

20  

22  20  19 


19 


22 


86 


FIRST  STEPS  AMONG  FIGURES. 


(26)       (27)        (28)        (29)        (30)       (31)      (32) 


2 

2               I 

3 

3 

I 

I               3 

I 

2 

3                X 

3 

3               9 

I 

I                I 

3 

3               3 

3' 

I                I 

3 

3                I 

2 

I                I 

3 

3              9 

9 

X 

3               I 

2 

3               3 

I 

2 

I                I 

2 

3 

3 

3          a 

3 

2 

2 

». 

I 

I           I 

— 

3 

2 

94 


21 


24 


91 


25 


26 


FIRST  STEPS  AMONG  FIGURES.  87 


(33) 

(34) 

(35) 

(36) 

(37) 

(38) 

(39) 

(40) 

I 

2 

I 

3 

I 

3 

I 

3 

3 

I 

2 

I 

2 

I 

3 

I 

3 

2 

I 

3 

I 

3 

I 

I 

2 

I 

I 

I 

2 

3 

I 

I 

I 

I 

2 

I 

3 

2 

I 

3 

3 

I 

I 

3 

3 

I 

3 

3 

I 

2 

2 

2 

I 

3 

3 

I 

I 

2 

3 

I 

3 

2 

I 

3 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

1 

2 

2 

2 

2 

3 

2 

I 

I 

I 

I 

I 

2 

I 

2 

I 

I 

I 

3 

3 

I 

2 

I 

2 

2 

I 

2 

3 

2 

I 

2 

I 

I 

3 

I 

I 

2 

3 

I 

2 

2 

3 

3 

2 

2 

I 

2 

I 

I 



3 

I 

I 



—    —    —     23      —     _     _     23 

26      24      23  27      27      25 

If  the  teacher  is  careful  that  the  pupils  do 
not  keep  any  of  the  solutions  of  the  foregoing 
examples,  they  may  be  given  2  or  3  times 
over,  first  solving  them  all,  then  solving  them 
all  again, and  so  on. 


88  FIRST  STEPS  AMONG  FIGURES. 


41)  (42)  (43)  (44)  (45)  (46)  (47)  (48) 

(49) 

«   23 

30 

13 

32 

23 

32 

33 

13 

3   32 

23 

21 

13 

33 

20 

21 

21 

1     12 

31 

33 

32 

30 

33 

13 

33 

2    21 

12 

10 

21 

12 

23 

23 

23 

J    30 

23 

32 

33 

33 

31 

32 

32 

3   12 

32 

13 

20 

22 

23 

33 

U 

3   3» 

33 

22 

32 

31 

32 

22 

22 

2  23         21         31         23         13  II  12  31 

3  33   20   23   33  s^      3^      31  20 
20  217  225  198  239  230  238  220  208 

(so)  ($■)  (5^)  (S3)  (S4)  (SS)  (S6)  (S7) 


32 

21 

31 

23 

30 

23 

33 

23 

21 

33 

23 

31 

23 

31 

21 

3f 

33 

20 

32 

33 

10 

23 

32 

23 

23 

13 

20 

30 

22 

30 

13 

32 

12 

32 

13 

22 

31 

22 

30 

33 

31 

33 

31 

13 

33 

13 

^^ 

12 

33 

21 

23 

31 

23 

32 

33 

31 

23 

13 

32 

23 

12 

21 

21 

23 

30 

32 

13 

30 

33 

33 

13 

32 

23 

31 

23 

23 

22 

23 

32 

33 

13 

33 

33 

33 

32 

32 

33 

23 

274  282  274   292  271  283  283  296 
See  P.  Ed.,  p.  33. 


FIRST  STEPS  AMONG  FIGURES.  89 


(S8)  (59)  (60)  (6.)  (6--)  ;63)  (64)  (65) 

(66) 

21 

33 

23 

3' 

33 

»3 

32 

21 

33 

33 

23 

31 

23 

21 

22 

13 

33 

22 

20 

31 

13 

33 

13 

31 

31 

32 

31 

'3 

20 

32 

20 

30 

33 

23 

23 

23 

32 

32 

.20 

13 

23 

21 

33 

30 

31 

23 

13 

33 

32 

32 

13 

21 

22 

12 

32 

33 

2 1 

23 

II 

32 

13 

33 

23 

21 

21 

13 

12 

33 

33 

10 

12 

32 

13 

32 

32 

22 

23 

20 

32 

32 

3» 

32 

12 

22 

33 

32 

13 

31 

23 

23 

23 

33 

30 

30 

33 

32 

23 

31 

13 

31 

21 

U 

21 

21 

23 

32 

13 

33 

12 

23 

32 

33 

33 

31 

31 

23 

32 

306    327    315    326    338   317  325  328  339 

Teach  pupils  to  prove  every  example  in  ad- 
dition by  adding  both  upward  and  downward, 
and  in  this  way  they  will  get  more  practice — 
just  what  is  needed. 

If  the  pupils  have  their  books — P.  Ed. — the 
following  examples  are  intended  to  be  given  at 
recitation  for  immediate  solution,  while  those 
in  the  P.  Ed.  may  be  solved  by  the  pupils  at 
their  seats  and  brought  to  recitation. 


90  FIRST  STEPS  AMONG  FIGURES. 

If  the  pupils  have  not  got  their  books  these 
examples  may  be  written  on  the  board  or  reacJ 
to  pupils  to  solve  at  their  seats,  a  few  daily : 

(>)        (2)         (3)         (4)         (5)         (6) 


231 

321 

333 

331 

^32 

135 

322 

233 

223 

223 

223 

332 

122 

213 

3'2 

"3 

331 

125 

333 

332 

233 

322 

233 

322 

212 

322 

213 

232 

213 

331 

323 

123 

322 

13' 

332 

215 

332 

332 

132 

303 

213 

321 

221 

233 

323 

233 

233 

33 

2,096 

2,109 

2,091 

1,888 

1,910 

i,8c8 

(7) 

^  (8) 

(9) 

(10) 

(") 

(12) 

33 

313 

213 

123 

221 

^31 

203 

232 

2 

331 

323 

333 

332 

231 

321 

322 

332 

123 

123 

323 

233 

233 

121 

232 

333 

333 

332 

123 

203 

313 

212 

232 

23 

312 

323 

233 

332 

321 

231 

233 

232 

321 

323 

233 

133 

331 

^23 

335 

223 

133 

213 

223 

333 

231 

331 

312 

321 

312 

312 

312 

2,445    2,663    2,022    2,543    2,523   2,663 

Review  these  if  need  be;  in  any  case  be  sure 
the  pupils  can  add  such  examples  as  the  above 
readily  and  accurately. 

See  P.  Ed.,  p.  36. 


FIRST  STEPS  AMONG  FIGURES.  9 1 


Examples  in  subtraction. 

13-  69758  M-  97856   15-  69587   16.  68059 
6035     74230     20152      604a 


63.723     23,626    49»435     62,019 

[7.  75860  18.  79685   19.  58796   2C.  96807 
2330     4242     25062      4200 


73»530    75^443    33,734     92,607 

Multiplication. 

21.  32032     22.  23103      23.  24130      24.  31402 
2322 


64,064  69,309  48,260  62,804 

25.    23103     26.  14023     27.  32023     28.  20312 
2  2  3  3 


46,206  28,046  96,069  60,936 

29.  40312  30.  31203 
2  3 

80,624  93.609 


^2  FIRST  STEPS  AMONG  FIGURES. 

Division. 

31.  2)48206   32.  3)90396   33.  3)69306 

24.103  30,132  23»I02 


34.  2)28460 

35-  3)39069 

36.  '. 

2)60482 

I4»: 

230 

13,023 

30.241 

37.  2)84^ 

)02 

38.  3)30960 

39-  : 

5)93600 

42,301 

10,320 

31,200 

(0 

(2) 

(3) 

(4) 

(5) 

(6) 

324 

434 

23 

43 

342 

431 

431 

431 

431 

324 

434 

343 

243 

444 

344 

442 

44 

242 

432 

213 

244 

243 

321 

434 

444 

332 

432 

134 

443 

244 

342 

441 

421 

424 

2 

42  1 

232 

424 

444 

^33 

3H 

314 

2^48     2,719     2,339     I '^43      i»90o     2,429 
See   P.  Ed.,  p.  38. 


FIRST  STEPS  AMONG  FIGURES.  93 


(7) 

(8) 

(9) 

(10) 

00 

(12) 

(i3> 

424 

31 

422 

431 

21 

34 

414 

434 

42 

344 

323 

44 

44 

445 

342 

44 

142 

444 

14 

21 

302 

241 

4 

433 

444 

23 

30 

244 

424 

31 

424 

123 

42 

44 

431 

332 

43 

344 

334 

44 

23 

444 

2,197 

^95 

2,109 

2,099 

188 

196 

2,27* 

(14) 

(15) 

(16) 

07) 

(18) 

(19) 

(20) 

214 

44 

44 

342 

244 

21 

23 

424 

213 

23 

444 

434 

34 

41 

444 

321 

32 

32 

423 

43 

34 

123 

444 

41 

421 

342 

21 

44 

434 

3 

44 

344 

231 

44 

3' 

341 

432 

31 

432 

444 

32 

23 

344 

424 

22 

444 

234 

43 

42 

2,324 

1 ,88 1 

237 

2,459 

2,352 

238 

238 

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 

(27) 

444 

21 

423 

432 

34 

442 

24 

344 

43 

444 

444 

42 

434 

43 

213 

24 

144 

324 

33 

324 

21 

422 

44 

231 

131 

23 

413 

4 

343 

32 

432 

423 

44 

32 

32 

214 

24 

344 

442 

3 

424 

20 

444 

33 

432 

343 

22 

321 

44 

321 

44 

123 

414 

31 

444 

23 

2,745   265  2,573  2,953   232  2,834  2 


94  FIRST  STEPS  AMONG  FIGURES. 


(28) 

(29) 

(30) 

(31) 

(32) 

{33) 

(34) 

243 

21 

44 

423 

341 

44 

34 

444 

432 

21 

341 

444 

21 

42 

231 

344 

34 

444 

34 

4 

44 

422 

211 

43 

332 

432 

44 

13 

444 

321 

43 

213 

121 

32 

22 

233 

432 

31 

441 

343 

12 

41 

423 

343 

24 

324 

444 

33 

44 

344 

444 

32 

432 

434 

14 

34 

212 

323 

43 

344 

232 

43 

31 

440 

342 

44 

443 

341 

44 

42 

3,436  3,213   359  3,737  3,166   291   347 


(35)  (36)  (37)   (38)  (39)   (40)  (41) 
343   423    14   314   232   21   32 


444 

442 

3 

432 

443 

42 

443 

321 

341 

41 

341 

444 

43 

344 

432 

234 

34 

134 

324 

14 

214 

344 

434 

42 

244 

431 

44 

423 

344 

341 

44 

441 

343 

21 

4 

234 

423 

34 

342 

234 

13 

234 

421 

412 

31 

444 

444 

44 

342 

432 

344 

23 

233 

323 

42 

444 

444 

431 

42 

324 

242 

33 

431 

3»759  3^825   308  3,249  3,460   317  2,9 

See  P.  Ed.,  p.  42 


II 


oiiii^'iE^.. 


FIRST  STEPS  AMONG  FIGURES. 


95 


(42)   (43)  (44)   (45)  (46) 


432 

343 
231 
424 
344 

243 
412 
424 

243 
432 
234 


34 

41 
4 

23 
42 

13 

24 
44 

33 
21 

43 


341 
423 
444 
34 
123 

442 
344 
431 
233 
342 
444 


24 
43 
32 
44 
4 
33 
24 
41 
43 
34 
42 


234 

421 

444 
324 
413 
442 
134 
224 
442 

431 
344 


(47)  (48) 

32  231 

44  443 

43  422 


24 
31 
44 
43 
24 
31 
44 
13 


334 
244 
413 
423 
341 
444 
413 
244 


3,762      322    3,601      364  3,853       373  3,952 

For  addition  and  multiplication.  (7  and  rev.) 

e 

7  3  5 


47  3  5 
345  6 


b 

47  3 
456 

f 
2647 
5673 
For  subtraction, 
b 


c 

264 

34  5 


II  9  8  12 
6745 


10  5  9 
436 


d 

3  5  2 

7  3  4 

h 
665 

7  3  7 


7  3 


10  7 
S3 


II  8 
45 


12  9  13  8 
7376 


c 
[069  7 

6345 

g 
9  8  10 

53    7 


12 
6 


107 
34 


13 
6 


96 


FIRST  STEPS  AMONG  FIGURES. 


For  division. 


a 

12 

2,S 

12  21 

6 

5 
d 

4  3 

i8 

35 

158 

3 

7 

34 

28  36  49  24 
7676 


30  21  42  6 
5    7'  63 


9  20  10  42 

3    4    5    7 
f 
24  20  18  35 
•4565 


g 
16  14  12 

4    7    3 


h 
28  15  30 
456 


(I)  (2)  (3)  (4)  (5) 


342 

24 

243 

404 

23 

421 

32 

434 

34 

34 

34 

44 

241 

423 

44 

z 

13 

343 

342 

3 

40  • 

234 

424 

43 

423 

34 

413 

334 

21 

342 

21 

324 

41 

34 

244 

44 

34 

423 

44 

321 

32 

443 

342 

32 

143 

42 

3 

333 

43 

432 

33 

234 

423 

4 

341 

4 

444 

342 

44 

132 

13 

321 

244 

32 

324 

44 

243 

344 

43 

431 

32 

342 

321 

32 

4,604         452      4,296      4,774        476 
See  P.  Ed.,  p.  40. 


FIRST  STEPS  AMONG  FIGURES.  97 

6.  Add  342,  234,  344,  421,  342,  24,  431, 
231,  4,  423,  344,  434,  324. 

7.  Add  24,  43,  24,  32,  44,  34,  42,  4,  34, 
2i,43»  14,  34,  23,43. 

8.  Add  43,  233,  424,  341,  3,  434,  4,  342, 
434,  342,  243,  414. 

9.  Add  2,  34,  44,  41,  32,  43,  34,  3,  44,  34, 
23»  42,  34,  41. 

How  many  quarts  are  there  in  three  gallons? 

Solution  :  In  one  gallon  there  are  4  quarts, 
in  three  gallons  there  are  three  times  4  quarts 
or  12  quarts. 

The  following  examples  may  be  given  during 
recitation  : 

1.  How  many  feet  have  5  horses? 

2.  Arthur  was  paid  5  cents  for  doing  an 
errand  and  his  sister  gave  him  4  cents  ;  how 
many  had  he  then  ? 

3.  If  a  carpenter  can  drive  3  nails  in  a 
minute,  how  many  minutes  will  it  tiake  him  to 
drive  18  nails  ? 

4.  Charles  had  10  snow  balls  in  a  pile  ;  he 
threw  4  of  them  at  his  playmates.  How  many 
remained  in  the  pile? 

7 


98  FIRST  STEPS  AMONG  FIGURES. 

5.  Mr.  Smith  paid  4  dollars  for  the  cloth 
for  a  pair  of  pants  and  2  dollars  for  making 
them  ;  what  did  the  pants  cost  him  ? 

6.  How  many  vests  at  $4  each  can  be 
bought  for  $24? 

7.  At  4  shillings  a  pair,  what  cost  5  pairs 
of  scissors  ? 

8.  If  a  hat  cost  $3  and  a  pair  of  boots  $10,' 
how  much  more  do  the  boots  cost  than  the  hat  ? 

9.  Andrew  has  a  pair  of  ponies,  how  many 
feet  have  they  ? 

10.  How  many  more  feet  than  eyes  has  a 
four-horse  team  ? 

11.  How  many  less  heads  than  feet  has  a 
three-horse  team  ? 

1 2.  A  boy  spent  2 1  cents  for   marbles  at  3 
cents  each  ;   how  many  marbles  did  he  get  ? 

13.  Barton  has  17  cents  ;  how  many  pencil^ 
at  2  cents  each  can  he  buy  and  keep  3  cents? 

3+5+4+5+5+3+4+5+3+4+2+3 , 
-+-2+4=  ?     Ans.  52. 

5+5+3+3+4+4+5+3+2+4+1+5 
+  4  +  5+3+4=?     Ans.  60. 
Examples  like  the  above,   having  only  addi 
tion,  may    be  given  both   forward  and  back- 
ward, thus  they  will  make  4  examples  instead 
See  P.  Ed.,  p.  45. 


FIRST  STEPS  AMONG  FIGURES.  99 

of  2.  Still  more  may  be  made  by  commenc- 
ing with  another  number,  as  12,  and  adding  12 
to  the  answer.     Thus : 

12  +  (5+5+3+3+4+4+5  +  3  +  2-1-4+ 
i  +  5+4+5+3+4)=6o+ 12=72. 
8  +  4  +  5+4^3x5+4  +  4+5-3-4-5 
—4—5—3—3^     Ans.  21. 

15  +  4  +  4-5-^3  +  5  +  5+4^4  +  7  +  3+4 

+  5+5?     Ans.  29. 

4+5+3+4+2+5+4+5+3+2+5+5 
+  4  +  4  +  3?  Ans.  58. 

3+4+5+3+1+5+4+5+2+3+5+4 
+  5+4  +  4  +  5?  Ans.  62. 

8  +  5+4  +  5  +  2  +  4+4x3—5-^4x8—4 
-5-5-4?  Ans.  14. 

16  +  4  +  5  +  3  +  4-4-7x6  +  5-3-5- 

4  +  2  +  5+6  +  5?  Ans.  9. 

18  +  5+4  +  4  +  5-3-4-5^4  +  5+5+- 
4x5+4  +  4+4?  Ans.  7. 

3  +  2+4  +  5  +  5-^3+4  +  4  +  2+5+3  +  4 
+  5+3  +  3  +  2+4?  Ans.  61. 

4+5+4+3^4x7+5-2-3-4-5-3 
—5-4x3  +  3+6?  Ans.  4. 

4+2  +  5  +  3+3  +  5+4  +  4  +  5+5+3-1-2 
+  4+1+5+4  I  5  +  3?  Ans.  67. 

3+4+4+5+2+5+3+3+4+5+4+4 
+  5  ^5+5+3  +  4?  Ans.  68. 


lOO  FIRST  STEPS  AMONG  FIGURES. 

3+4+5+3+4+5+5+4+4+3+5+2 

+  5  +  3  +  5+4  +  5?     Ans.  69. 
5-r4  +  5+4  +  5  +  i-6  ^-3 +  5 +4  + 5-3 
—5  —  1-1-4x7—5—4?     Ans.  12. 

7+4+5+5+4+4+3—5-4-5-3-4 
—4  X3?     Ans.  21. 

5  +  3  +  5  +  4  +  4  +  3  +  3  +  5  +  5-3-37-4 
-5-5-4-4-3  •     Ans.  6. 


(I) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

35 

5 

54 

50 

34 

45 

32 

54 

43 

35 

35 

54 

53 

45 

5 

54 

43 

4 

45 

24 

53 

43 

35 

5 

53 

33 

45 

34 

24 

42 

52 

45 

54 

51 

15 

55 

5^ 

44 

24 

44 

35 

52 

34 

35 

25 

55 

32 

54 

41 

52 

54 

53 

31 

55 

5 

35 

45 

45 

21 

44 

34 

32 

54 

347     364     332     341     385     344     361 
Examples  in  subtraction : 
I.  67,548      2.  69.584      3.  75,897      4.  97^867 
43^235  34.331  54.353  52,343 


24,313  34,253  21,544  45*524 

5.  64,786      6.  79,684 
2,432  7,053 


62,354  72,631 

See  P.  Ed.,  p.  47. 


FIRST  STEPS  AMONG  FIGURES.  lOI 

The  teacher  may  use  his  own  judgmeiii.  as  to 
teaching  subtraction  when  some  figures  of  the 
subtrahend  are  greater  than  the  corresponding 
figures  of  the  minuend. 

A  purely  mechanical  method  is  here  given 
with  the  idea  that  the  method  of  doing  many 
things  may  properly  precede  the  reason  for  the 
method. 

If  the  following  method  be  used,  after  one 
or  two  years  or  in  a  larger  book  the  reason  of 
the  method  should  be  fully  explained  to  the 
pupil,  and  he  should  then  be  required  to  give 
the  reasoning  himself. 

32,413  —  5,667.  Solve  by  separating  the 
figures  of  the  minuend,  as  in  the  line  below, 
and  then  when  any  figure  of  the  subtrahend  is 
smaller  that  the  figure  above  it,  write  i  before 

3  I  2   I  4  I  I    I  3 
it  thus  :  5667  and  then  subtract, 

26746 
being  careful  when  i  is  prefixed  to  the  upper 
figure  to  add  i  to  the  next  left  hand  figure  of 
the  subtrahend.  The  following  examples  are 
so  arranged  that  no  figure  of  the  subtrahend 
is  greater  than  7,  the  tables  having  been 
learned  only  so  far.  After  solving  the  above 
example  the  pupil  should  say  5,667  from  32,413 
leaves  26,746. 


I02  FIRST  STEPS  AMONG  FIGURES. 

7.  910201    8.  423423    9-  621423 

56454  45466  53667 

853747  377957  567756 

As  soon  as  the  pupil  understands  the  mechan- 
ical work,  he  should  not  be  allowed  to  write  the 
I's  in  the  minuend,  but  imagine  them  to  be 
there. 

10.  831,242      II.  513-423      12.  430^213 
63.567  46,647  262,637 

767,675  466,776  167,576- 


13-  731,420      14.  342,031      15.  532,514- 
54,654  25,266  65,251 


676,766            316,765  467,263. 

16.  731,420      17.  624,091  18.  831,042 

61,265              62,035  63.415 

670,155             562,056  767,621 
See  P.  Ed.,  p.  48. 


FIRST  STEPS  AMONG  FIGURES.  1 05 


(0 

532/ 

(3) 

(4) 

(5) 

(6) 

24 

523 

24 

355 

544 

55 

455 

454 

55 

432 

352- 

32 

44 

342 

44 

534 

435 

44 

503 

535 

32 

255 

534 

53 

434 

453 

53 

543 

253 

25 

355 

544 

45 

424 

545. 

43 

3 

325 

54 

354 

324 

5 

545 

454 

34 

535 

432 

54 

334 

543 

43 

444 

543 

335      3.205     4,173        384      3,876      3,962 

More  examples  may  be  made  from  these  by- 
reading  tHem  from  the  center  each  way,thus 
giving  new  combinations,  or  by  giving  two  ad- 
ditional numbers,  one  above  the  upper  number 
and  one  below  the  lower  one ;  in  this  way  the 
combinations  will  be  different  whether  the 
pupil  add  upward  or  downward. 

Of  course  the  teacher  must  add  the  sum  of 
these  two  numbers  to  the  answer  in  the  book  to 
get  the  answers  of  the  new  example.  Exam- 
ples in  Pupils'  Edition  may  be  treated  in  the 
same  way. 

7.  30,142      8.  23,103      9.  42.301     10.  24,130 
2322 

60,284  69,309  84,602  48,260 


I04  FIRST  STEPS  AMONG  FIGURES. 

Show  the  pupils  that  in  multiplying,  as  in 
adding,  if  any  result  is  greater  than  9,  the  left 
hand  figure  is  added  to  the  next  result,  which 
is  of  the  same  kind. 


II.  63,524    12.  36,546    13.  36,426     14.  63,524 


3 

2        3 

4 

190,572 

73,092    109,278 

254,096 

IS-  26,463 

4 

16.  64,524 

4 

17.  53»625 
5 

105,852 

258,096 

268,125 

18.  26,463 

5 

19-  46,035 
4 

20.  26,304 
6 

132,315 

184,140 

157.824 

21. 

25.036 

4 

22.  50,264 
6 

100,144  301.584 

Caution  :  Do  not  allow  the  pupil  to  write  any^ 
wher€  what  he  is  to  add  to  the  next  product. 
See  P.  Ed.,  p.  52. 


FIRST  STEPS  AMONG   FIGURES. 


105 


For  addition  and  multiplication.  (8  and  rev.) 

alb         c     I     d     I     e     I     f     I     g 
58473  62584  73625 !84736!2s847j36258|47362 
3456718345678345167834156783145678134567 
h       i  I 
5847  362 
8345  678I 


For  subtraction. 


a 

b            1 

c 

9   13   10   »4  5 
56783 

9   13  8   12  6     9   II    14  10 
453443876 

14 
6 

d 

e             •            f 

II    15   10  6   10 
78534 

7   II    i;    12   139   12  8   II 
5     6     7     8     7I6     5  4     3 

13 
8 

g 
10  7   II  8  I 

3456 

h         i       i 
2  16  7   II  8112  9  10 
783     4  5!  6  7     8 

For  division. 

a 

24  49  24  18  8 

67834 

b            1            c 
40  20  6  48  21I42  20  32  15 
5     438     7i  6    5    4    3 

1 

d             1             e 
9  24   10  30  56132  28  12  6435 
3456     7I  84387 

f 
12  30  12  21 
6543 

14 
7 

36   15  35   16  24 
65543 

h 
40  18  42  15 
8675 

i 
[6  48  28 
867 

Io6  FIRST  STEPS  AMONG  FIGURES. 


(24) 

(25)  (26) 

(27) 

(28) 

(29) 

345 

235 

25 

435 

345 

5+ 

234 

543 

54 

543 

523 

4S 

532 

455 

43 

234 

454 

25 

424 

345 

55 

455 

325 

54 

352 

432 

42 

543 

543 

24 

443 

554 

21 

235 

454 

55 

345 

345 

54 

352 

535 

32 

532 

553 

35 

544 

234 

54 

314 

345 

54 

535 

555 

43 

3,521    3,807  383    3,876    3,968      38+ 


(30)   (30  (32)   (33)   (34)   (35> 


545 

435 

45 

543 

231 

4S 

434 

554 

4 

355 

545 

53- 

55 

345 

23 

234 

354 

4+ 

53 

534 

54 

542 

434 

53 

224 

242 

45 

345 

543 

2S- 

532 

555 

33 

554 

355 

44 

454 

343 

54 

435 

332 

2S 

534 

543 

45 

543 

435 

5+ 

325 

254 

54 

343 

543 

45 

43 

442 

33 

555 

345 

55 

3,199 

4,247 

390 

4,449 

4,117 

439 

See  P.  Ed.,  p.  59. 

FIRST  STEPS  AMONG  FIGURES. 


I0> 


Tlie  following  tables  are  an  excellent  prepa- 
jation  for  short  division. 

Before  the  pupils  solve  the  examples  on  page 
120,  give  them  a  review  of  this  page. 

Division  with  remainders. 

*5's  (and  review.) 

a  I         b  c        I       d        I  e 

i6  lo  3  ii\22  9  6  13  5   14  4  718  13  II 
3     42     3,  424     32     432I4     3     2! 

6's  (and  review.) 

a         1         b         j  eld 

14  16  3  26  II   26  II  921   10  19  733  14  9  5 

4326:54361543215623 


II  22  18  7 

2454 


3  5  38  27 


g      I       h 
20  13  8  19  7  32 


326562  3!  456 


1.  What  cost  8  dozen  buttons  at  7  cents  a 
dozen  ? 

2.  What  cost  a  pair  of  boots  at  $7  and  a 
hat  at  $5  ? 

3.  How  much  more  does  a  reader  cost  at  & 
shillings  than  a  speller  at  2  shilli.igs? 

♦These  are  to  be  recited  as  follows  :   3  in  1 6,  5  times  and 
I  rapiainder,  &c. 


I08  FIRST  STEPS  AMONG  FIGURES. 

4.  There  were  9  birds  in  a  flock,  and  a 
hunter  killed  all  but  4  ;  how  many  did  he  kill  ? 

5.  How  many  knives  at  7  shilling's  each 
may  be  bought  for  35  shillings  ? 

6.  A  boy  spent  21  cents  for  marbles  at  3 
cents  each  ;  how  many  marbles  did  he  buy? 

7.  There  are  8  pigs  in  one  pen  and  5  in 
another  ;   how  many  in  both  pens  ? 

8.  A  boy  earned  8  cents  on  Monday,  7 
cents  on  Tuesday,  and  6  cents  on  Wednesday  ; 
how  many  cents  did  he  earn  in  the  3  days  ? 

9.  Henry  bought  8  marbles  it  4  cents 
«ach  ;  what  did  they  cost  him  ? 

10.  13  boys  were  skating  on  a  pond  ;  during 
the  afternoon  8  of  them  fell  upon  the  ice.  How 
many  of  them  did  not  fall  ? 

11.  Fred  had  15  cents,  he  spent  5  cents  for 
oranges  and  i  cent  for  candy ;  how  many 
cents  had  he  left  ? 

12.  Georije  had  a  bank  into  which  he  put  7 
cents,  his  father  8,  and  his  sister  4  ;  how 
many  cents  had  he  in  his  bank  ? 

13.  Charles  has  6  cents  and  his  sister  has  2 
cents  more  than  he ;  how  many  cents  have 
both? 

14.  Henry  had  $8  for  Christmas  and  his 
sister  half  as  many  ;  how  many  had  both  1 

See  P.  Ed.,  p.  56. 


FIRST  STEPS  AMONG  FIGURES.  IO9 

15.  William  bought  an  orange  for  4  cents,  a 
fig  for  I  cent  and  some  candy  for  2  cents.  He 
sold  them  all  for  12  cents.  How  much  did  he 
gain? 

16.  A  boy  received  6  cents  a  bushel  for 
picking  hops  ;  he  earned  in  this  way  48  cents. 
in  one  day.     How  many  bushels  did  he  pick  ? 

17.  Carrie  whispered  3  times  in  i  day,  for 
each  time  she  whispered  she  had  to  remain 
after  school  5  minutes  ;  how  long  did  she  have 
to  remain  ? 

18.  20  cents  are  to  be  divided  equally 
among  5  boys  ;  how  many  cents  should  eacb 
boy  receive  ? 

19.  If  6  pieces  of  tape  cost  24  cents,  how 
much  did  one  piece  cost  ? 

20.  If  a  boy  earned  28  shillings  in  7  days, 
how  much  did  he  earn  in  i  day  ? 

21.  Samuel  walked  28  miles  in  4  days  ;  at 
that  rate  how  far  would  he  walk  in  i  day  ? 

22.  How  many  quarts  of  milk  at  6  cents  a 
quart  can  be  bought  for  36  cents  ? 

23.  If  4  gallons  of  molasses  cost  28  shil- 
lings, what  cost  I  gallon  ? 

24.  56  cents  will  hire  how  many  boys  for  an 
hour,  if  each  boy  is  to  have  7  cents  for  an 
hour's  work  ? 


110  FIRST  STEPS  AMONG  FIGURES. 


(0 

(2) 

(3) 

(4^1 

(5) 

(6) 

(7) 

52 

242 

543 

1 2 

34 

535 

344 

45 

525 

244 

54 

53 

454 

553 

31 

343 

554 

35 

4 

342 

435 

53 

451 

343 

43 

25 

535 

234 

44 

535 

234 

51 

40 

453 

544 

25 

434 

455 

45 

32 

341 

345 

53 

453 

542 

34 

44 

355 

453 

44 

345 

435 

34 

53 

434 

423 

35 

523 

252 

23 

45 

544 

345 

34 

254 

344 

45 

44 

355 

534 

55 

535 

535 

53 

35 

433 

453 

471  4,640  4,481     459     409  4,781  4,663 
S6 
(11;     (12)     (13)     (14)      (15)     (16) 


544 

543 

443 

532 

345 

432 

354 

441 

355 

344 

53 

544 

435 

355 

534 

434 

434 

434 

543 

434 

433 

453 

545 

542 

431 

542 

341 

345 

354 

344 

303 

344 

432 

524 

534 

453 

454 

354 

544 

343 

435 

341 

435 

425 

342 

234 

543 

535 

544 

543 

455 

555 

345 

344 

432 

254 

544 

443 

254 

455 

435 

434 

345 

344 

342 

532 

353 

543 

432 

535 

545 

344 

5,263  5^12  5,200  s,o86  4,729  5,300 
See  P.  Ed.,  p.  59. 


FIRST  STEPS  AMONG  FIGURES. 


Ill 


Pupils  read  : 

i.  3040321    2. 

30245000 

3- 

463317030 

4.  500030261   5. 

2000032 

6 

15000000 

7.  320030000  8. 

674346537 

9- 

42000 

10.  400320219  II. 

3605000 

12. 

50000018 

13.  463308260  14. 

75000341 

^5- 

lOIOOIO 

Read  the  following : 

..  67345768       2.  476347854  3.  74000037 

4-  735400005     5.  900007000  6.  3 16000 1 40 

7.  80370000       8.  700000004  9.  7020500 

lo,  86000045   II.  800006000  12.  90000007 

13.  8060700     14.  90430000  15.  735468371 


Teach  pupils  to  write  Arabic  to  billions,  that 
is  including  999,999,999.  Teach  the  pupils  to 
numerate  by  periods  to  the  right  as  well  as  to 
the  left.  Thus:  units,  thousands,  millions; 
millions,  thousands,  units,  until  they  are  per- 
fectly familiar  with  it. 

Method:  Suppose  the  number  ten  million 
ninety  thousand  three  is  to  -be  written.  Instruct 
the  pupil  to  write  the  number  of  millions  first 
with  a  comma  after  it,  and  that  the  first  period 
at  the  left  does  not  need  to  be  filled  to  three 
4)laces  by  prefixing  ciphers.      For  the  above 


lit  FIRST  STEPS  AMONG    FIGURES. 

number  the  pupil  will  write  lo,  at  first.  Teach- 
er ask  lo  what?  Pupil,  lo  million.  Teacher; 
What  period  is  next  to  right  of  millions  ?  Pu- 
pil :  thousands.  Teacher :  How  many  thou- 
sands are  there  (in  this  number)  ?  Pupil  : 
ninety.  Teacher  :  Write  it  after  the  comma, 
and,  as  it  fills  but  two  places,  place  a  cipher  at 
the  left  of  the  90  and  a  comma  after  the  90. 
The  number  will  now  be  10,090.  Teacher: 
You  have  now  millions  and  thousands ;  what 
period  is  next  ?  Pupil :  units.  Teacher :  How 
many  units  are  there  in  this  number?  Pupil : 
three.  Teacher  :  Place  the  3  to  the  right  and 
prefix  two  ciphers  to  it  to  fill  the  three  places 
of  the  periods.  Place  a  period  at  the  right 
because  it  is  the  end  of  the  number,  and  you 
have  10,  090,  003. 

Teach  the  pupil  when  writing  numbers  at  the 
blackboard  to  turn  directly  away  from  it  as 
soon  as  units  and  the  period  are  written,  for  he 
should  be  sure  that  the  number  is  correct  with- 
out numerating  to  the  left. 

Teach  the  pupils  of  course  when  there  are 
no  thousands,  to  write  three  ciphers  and  treat 
unit's  period  in  the  same  way. 

When  teaching  to  write  billions,  trillions, 
etc.,  follow  the  same  method. 


FIRST  STEPS  AMONG  FIGURES.  I  13 

Teach  the  pupils  that  for  the  word  hundred, 
you  will  write  on  the  board  hun. ;  for  thousand, 
th.  ;  for  million,  mil.  ;  and  when  you  get  so 
far,  for  billion,  bil. ;  for  trillion,  tr.  ;  for  quad- 
rillion, quad.,  etc.  This  method  will  save  the 
teacher  much  labor  and  much  space  on  the 
blackboard.  Thus  the  teacher  may  write  upon 
the  board : 

"  Write  in  Arabic  five  mil.  forty  th.  six." 
To  be  written  on  the  blackboard  for  pupils 
to  bring  to  recitation  written  in  Arabic : 

1.  Write  in  Arabic  four  th.  fifteen. 

2.  "  "       twenty  mil.    three  hun. 

3.  "  "       nine  mil.  forty  th. 
Caution  :  Teach  pupils  to  put  a  comma  only 

after   each  period^   except  the  last.     Thus  in 

Note. — The  following  diagram  may  assist  pupils  in  writing 
numbers,  but  after  being  used  a  few  weeks  the  pupils  should 
write  numbers  without  using  it. 


Millions. 
35 


Thousands. 


Units. 
003 


058 

The  teacher  may  draw  a  diagram  like  the  above  and  allow 
the  pupils  to  write  numbers  in  it,  as  the  number  35,058  003 
is  placed  there.  Teach  the  pupils  that  the  third  period 
represents  millions,  and  that  each  period  is  r»ad  as  if  it  stood 
alone,  only  that  its  name  is  given. 

8 


114  FIRST  STEPS  AMONG  FIGURES. 


three  hundred  seventy-five  million,  four  hun- 
dred eight  thousand  seven  hundred  forty,  the 
pupil  may  have  an  idea  that  he  should  put  a 
comma  for  hundreds,  whereas  the  above  num- 
ber should  be  written  374,408,740.  When  no 
name  is  given  to  a  number  it  is  supposed  to  be 
units ;  e.  ^.,  two  thousand  three  hundred  eight. 
Eight  here  means  eight  units,  and  three  hun- 
dred eight,  (which  has  n6  name  given  to  it)  is 
308  in  units  period. 

The  following  examples  may  be  written  upon 
the  board  and  the  pupils  required  to  bring  ^he 
answers  to  class : 

1.  Write    in   Arabic,   thirt}'   million,  eight 
thousand,  thre?  hundred  fifty-one. 

2.  Write  in  Arabic,  two  hundred  fifty  thou- 
sand. 

3.  Write   in  Arabic,  one  hundred    sixteen 
million  two  hundred  twenty. 

4.  Write  in  Arabic,  three  hundred  million, 
sixty  thousand,  five  hundred,  seven. 

5.  Write  in  Arabic,  five  hundred  thousand. 

6.  Write  in  Arabic,  one  million,  one  thou- 
sand, one. 

7.  ^^'rite    in    Arabic,  seventy    million,  six 
hundred  thousand,  eighty. 

See  P.  Ed.,  p.  61. 


FIRST  STEPS  AMONG  FIGURfeS.  i»5 

8.  Write  in  Arabic,  one  hundred  fifty-four 
million,  two  hundred  sixty-one  thousand,  five 
hundred  forty-eight. 

9.  Write  in  Arabic,  eight  hundred  million. 

10.  Write  in  Arabic,  three  million,  three. 

11.  write  in  Arabic,  ten  thousand,  tep. 

12.  Write  in  words,  809271300. 

13.  Write  in  Arabic,  five  hundred,  four 
million,  forty. 

14.  Write  in  Arabic,  ten  million,  ten  thou- 
sand. 

15.  Write  in  Arabic,  one  hundred  one 
million,  one  hundred  one. 

The  Roman  notation  uses  the  following 
letters:  I  =  i,  V  =  5.  X=io,  L=:5o.  C  = 
100,  D  =  5oo,  M  =  iooo. 

To  read  a  number  expressed  in  the  Roman 
notation  : 

*Rule :  Add  the  values  of  the  letters,  observ- 
ing that  when  a  letter  is  followed  by  one  of 
greater  value  than  itself,  the  difference  between 
the  two  is  to  be   taken   in  making  up  the  sum. 

16.  Write  in  Roman,  three  hundred  forty- 
five. 

17.  Write  in  Roman,  one  hundred  seventy- 
four. 

•  From  Olney's  Elements  of  Arithmetic. 


Il6  FIRST  STEPS  AMONG  FIGURES. 

18.  Write  in  Roman,  four  hundred  sixty-two. 

19.  Write  in  Roman,  six  hundred  ninety-six. 

20.  Write  in  Roman,  eight  hundred  ninety- 
nine. 

21.  Write  in  words,  245306341. 
22..  Write  in  words,  32743642. 

23.  Write  in  Roman,  three  hundred  eighty- 
seven. 

For  rapid  solving. 

1.  4-H3  +  5  +  2  +  6  +  4  +  5+6  +  3  +  5+4 

+  3^-6  +  6  +  5=?     Ans.  67. 

2.  3  +  5-f44-64-3  +  6  +  5  +  2  +  6  +  5-F5 

+  4  +  6  +  64-2  +  4=  ?     Ans.  72. 

3.  5  +  6  +  6  +  4  +  3  +  5  +  4  +  3  +  6  +  2+6 

+  6  +  5  +  6  +  2+4=1     Ans.  73. 

4.  4  +  3  +  6  +  5  +  2  +  6  +  6  +  5+4  +  3  +  6 

+  5  +  4  +  6  +  6  +  5=  ?     Ans.  76. 

5.  5  +  6  +  3  +  4  +  6  +  5  +  6  +  6  +  5+4  +  3 

+  6  +  6  +  6  +  5+4=?     Ans.  80. 

6.  15  +  6-^7x5  +  3-^3x8  +  3+5-^7x4 

+  5+3-5-6=?     Ans.  29. 

7.  I9  +  4+5-^4x6-6-5-6-^5x8-4 
-4-6  +  5  —  6  X  7—4=  ?     Ans.  31. 

8.  7x6  +  5  +  6  —  4—6  —  6  —  5-8x6  +  5 

+  6—3—6—5+6=  ?     Ans.  27. 

9.  6x8  +  6— 2-5-6-5-4-J-4X7-5 

—4—3  —  6  —  6+4=?     Ans.  8. 


FIRST  STEPS  AMONG  FIGURES.  II7 

10.  8  +  7+6  +  5  +  5  +-4-^-5x6-5-6-6 

-6-3-^8  +  5=?     Ans.  7. 

11.  53-6-5-6-4-5+6  +  5+4-^6x5 

—6  —  5  —  6  +  5=?     Ans.  23. 

12.  47+5—6—4—5-2-4-5x8  —  5-6—6 

+  3-7-6+8  +  6=  ?     Ans.  21. 
13-  23-5-6-4x6  +  5-6-5-=-7X5-6 
—6-6x8  —  6=1     Ans.  i8. 

14.  16-7-4x6  +  5—4  —  6  —  5  +  6  +  5+6+4 
-^7X 8— 4-4-5=.?     Ans.  27. 

15.  6x7-6  — 5-4— 3-T-3  +  5+6  +  6  +  4 

+  5-6  — "5- 6-2-^5=  ?     Ans.  3. 
i6.  28  +  6  +  6  +  5  +  4— 6—5— 6— 4-H4X5 
—6—5-7-8x7=?     Ans.  21. 

17.  61—6—6  —  5—5—4-6—3-6^5x8 

—  5  —  6-5-7  +  6=?     Ans.  9. 

18.  34-6-5-2^3x5-4-6  +  3^7  +  7 

+  5+5  +  6  +  6=  ?     Ans.  3^- 

19.  I6  +  5+4  +  3  +  5-6-3-^8x5-6  +  4 

+  5+5-6-4=^     Ans.  13. 
30.  27+5  +  4+-6X8-6-4-5-3-6-5 
—4-7-3x6-4—4-6=?    Ans.  id 

The  teacher  is  advised  to  give  a  few  exam- 
ples in  subtraction  each  day,  and  with  them  a 
few  in  multipHcation  and  perhaps  in  division 
also. 


Il8  FIRST  STEPS  AMONG  FIGURES. 

Subtraction. 

I.  3»423»056     2.  7.352»043     3-   9»635»024 
654.368  834,657  746,548 


2,768,688  6,517,386  8888,476 

4,  4,320,032    5.  63,140,052    6.  8,400,314 
543.054  6,572,036  50^248 


3.776,978         56,568,016        8,350,066 


7.  7,360,042  8.  61,420,035  9.  5,304,036 
2,500,075      540,257     202,356 


4.859,967    60,879,778    5,101,680 


10.    8.340050       II.    64,230,051        13.     75.310,040 

762,034       5^0'765       230,076 


7,578,016     63,719,286     75,079,964 

13-  6,343.520-656245-  ?  Ans.  5.687,275- 

M-  94,530,062-8,240,277  =  ?  Ans.  86,289,785. 
See  P.  Ed.,  p.  64. 


FIRST  STEPS  AMONG  FIGURES.  1 19 


•Multiplication. 

I.  648,057 
8 

2.  746,805 
7 

5^227,^35 

3.  470685 
6 

5.I84-456 

2,824,110 

4    8,640753 
7 

5-  680,574 
8 

6.  358,407 
6 

60,485,271 

5  444,592 

2,150,442 

7.  685,740x4=  ?     Ans.  2,742,960. 
8.7,406,8^3x7=?     Ans.  51.847971. 

The   teacher   should    solve   an    example  in 
which  there  are  two  figures  in  the  multiplier. 


6,354^23=?     Ans.  146,142. 
53,462x32=.?     Ans.  1,710,784. 
36,425  X  34=  ?     Ans.  1,238.450. 
353.625  X  43=  ?     Ans.  15.205,875. 
563.524  X  65  =  ?     Ans.  36,629,060. 
350,264  X  36=  ?     Ans.  12,609.504. 
526,304x34=.^     Ans.  17.894,336. 
640,536x64=?     Ans  40.994,304. 
4675^45=  -     Ans.  210,375. 


•The  numbers  used  in  these  examples  are  puinted  off  !n 
periods  for  convenience  in  copying  to  blackboard  or  slate. 


I20  FIRST  STEPS  AMONG  FIGURES. 

Before  taking  up  examples  in  short  division 
review  division  with  remainders,  p.  107. 

In  teaching  pupils  short  division  when  there 

are  remainders  during  the  operation,  write  the 

figures  of  the  dividend  well  apart ;  thus,  in  the 

example  379531 -^4,  ^fite  ♦>3  7 19  35  33  "      ^^^j 

94003^ 
write  the  remainder  before  the  next  figure  as  in 

the  example  given.  After  solving  2  or  3  exam- 
ples in  this  way,  write  the  figures  closely,  in  the 
usual  form  on  the  blackboard,  and  let  a  pupil 
divide  orally,  the  teacher  using  the  crayon,  one 
pupil  telling  how  many  times  it  is  contained 
and  what  remainder,  the  next  stating  what  the 
next  partial  dividend  is  and  how  many  times 
the  divisor  is  contained  and  what  remainder, 
etc. 

Let  the  pupils  first  solve  the  examples  with- 
out remainders,  given  in  Pupils'  Edition,  and 
the  following  4  examples  : 

1.  24129318-^3=  ?     Ans.  8,043,106. 

2.  281,683.220-7-4=?     Ans.  70,420,805. 
3.12,246,9214-3=?     Ans.  4,082,307. 

4.  322,412,836-^-4=?     Ans.  80,603,209, 
With  remainders. 

5.  83,923—3=  ?     Ans.  27,974>i. 

6.  182,539^4=  ?     Ans.  45'634^. 

See  P.  Ed.,  p.  71. 


FIRST  STEPS  AMONG  FIGURES.  121 

7.1,373,224-3=?     Ans.  457.74ri 
S-  9^354  3^8-^-4=  ?     Ans.  22,838,582. 
9.  273.78i-=-5=?     Ans.  54,756'. 

10.  1,035,879-5-6=  ?     Ans.  172.646'. 

11.  2,683,507-4-4=?     Ans.  670,876'. 

12.  3,921,278^6=  ?     Ans.  653,546V 
Method  of  teaching  pupils  to  add  numbers 

like  46  and  7. 

In  adding  46  and  7,  ask  the  pupil  what  he 
should  add  first,  and  either  get  from  him  or 
show  him  that  6  and  7  are  to  be  added  first; 
that  it  makes  13,  of  which  the  right  hand  figure 
is  3,  which  will  be  the  right  hand  figure  of 
the  sum  of  46  and  7,  and  that  46  and  7  are 
53.  Persevere  in  this  plan  upon  the  following 
numbers  or  until  the  pupil  in  adding  such  num- 
bers as  37  and  8,  will  say  at  once  *' the  right 
hand  (or  least)  figure  will  be  5  ;  37  and  8  are 
45-" 


45 +6. > 

67  +  5? 

58  +  7-'' 

65+8? 

86  +  5? 

58  +  6? 

34  +  7? 

57  +  5? 

26  +  8? 

65  +  7? 

37+8? 

78  +  5? 

86  +  7? 

37+4? 

584-8.^ 

63  +  7? 

47  +  7? 

28-H4-^ 

76  +  5? 

53+8? 

37  +  6? 

76  +  6.? 

67  +  a? 

38  +  7? 

64  +  8.^ 

35  +  5? 

47  +  6? 

28  +  3.? 

122 


FIRST  STEPS  AMONG    FIGURES. 


43  +  7 

64  +  4? 

45  +  8? 

26  +  5? 

47  +  2: 

38  +  6I 

53  +  3? 

74  +  7? 

45  +  4^ 

76  +  8? 

37  +  5? 

58  +  2? 

53  +  6: 

'         24  +  3? 

65  +  7? 

86  +  4? 

27  +  8: 

68  +  5? 

83  +  2? 

54  +  6? 

65  +  3: 

'         36  +  7? 

57  +  4? 

68  +  8? 

43  +  5-' 

54  +  2? 

35  +  6; 

16  +  3? 

47  +  7.' 

38  +  4? 

53  +  8? 

24  +  5^ 

45  +  2 

86  +  6? 

47  +  3? 

58  +  7?    ' 

43  +  4 

?         34  +  8? 

75  +  5? 

56  +  2? 

27  +  6? 

48-^3? 

The  exercise  above  should  be  rriost  used  un- 
til the  pupils  are  perfectly  familiar  with  the 
combinations  given,  which  embrace  all  between 
8  and  3  inclusive  and  some  are  given  twice. 
It  may  first  be  given  to  the  pupils  in  the  order 
above,  then  commence  in  the  middle  of  the  ex- 
ercise and  go  each  way.  It  should  not  be 
written  upon  the  blackboard  but  recited  orally 
from  the  reading  of  the  examples  to  the  pupils. 

1.  A  boy  bought  a  top  for  18  cents  and  sold 
it  so  as  to  gain  7  cents  ;  what  did  he  sell  it 
for? 

2.  James's  mother  gave  him  30  cents  with 
which  to  buy  oranges.  At  6  cents  each  how 
many  could  he  buy  ? 


FIRST  STEPS  AMONG  FIGURES.  1 23 

3.  Willie  said  he  had  4  cents  ;  John  said  he 
had  4  times  as  many  ;  how  many  had  John  ? 

4.  George  had  8  sticks  of  candy  and  his  sis- 
ter had  7  ;  how  many  did  both  have? 

5.  18  ripe  peaches  were  on  a  tree  and  a  bad 
boy  stole  7  of  them  ;  how  many  were  left  ? 

6.  A  flock  of  18  birds  lit  upon  the  ground  ;. 
a  hunter  shot  11  of  them;  how  many  were 
left?     Ans.  II. 

7.  How  many  fingers  have  4  boys  ? 

8.  Charlie's  mother  gave  him  7  cents,  and 
his  sister  gave  him  enough  to  make  13  cents  ; 
how  many  did  his  sister  give  him  ? 

9.  John  had  10  cents,  one  of  his  sisters  had 
7  cents  and  the  other  had  6 ;  how  many  cents 
did  both  the  sisters  have  ? 

10.  Mary  bought  2  yards  of  calico  for  a 
doll's  dress  ;  she  gave  8  cents  a  yard  ;  how 
much  did  the  dress  cost  ? 

11.  A  tired  school  teacher  struck  a  naughty 
boy  five  times  upon  each  hand  ;  how  many 
times  did  she  strike  him  ? 

For  addition  and  multiplication. 
9's  (and  review.) 


a      I      b 
85963    748';9 
45678 1 94567 


c     I      d      I     e     I     f     I     gr 

6374  j  85963 I74859I62748  59637 
8945  67894,56789,49678194567 

Sec  p.  Ed. ,  p.  74. 


124 


FIRST  STEPS  AMONG  FIGURES. 


h 

4859 
8945 


6374 
6789 


For  subtraction. 


II    16  ii  13 

8995 

d 

II   17   12   14 

4876 


8  15   13   18 
5789 


3   15    10   12 
7654 


9   15   7    '2 

5     9  4     5 

h 

10   14   13   II 

4     9     4     5 


II     16     14     12 
6789 

f 

10     16     II     13 

6876 


9   13   15   10 
6987 


.1 
12   14   14 

6     5     7 


k 
12   17  9 
894 


For  division. 


g 
28  42   18  24 

7694 


a 

b 

c 

42  54  25  32 
7654 

24  63   16  40 
8945 

30  63  48 
6     7     8 

d 

e 

f 

27  28  72  35 
9487 

48  20  54  12 
6594 

35  24  64 
5     6     8 

h 

81  40  56 

9     8     7 


45  3^  30 
9     4     5 


FIRST  STEPS  AMONG  FIGURES. 


25 


j 

k 

1 

18  49 

32  72 

18  36  56 

2 

I  36  45 

6  7 

8  9 

298 

7  6  5 

(I) 

(2) 

(3)    (4) 

(5) 

36 

654 

245 

+5 

456 

53 

343 

634 

56 

546 

46 

565 

453 

+3 

323 

52 

654 

546 

H 

654 

64 

432 

635 

55 

65 

45 

543 

364 

6 

556 

34 

56 

655 

+6 

343 

65 

435 

536 

55 

656 

395 

3,682 

4,068    370 

3o99 

(6) 

(7) 

(8)    (9) 

(10) 

65 

356 

654 

0 

+5 

256 

6 

645 

565 

56 

643 

35 

546 

634 

+3 

345 

64 

356 

362 

56 

656 

23 

465 

556    i 

55 

562 

46 

321 

665 

26 

234 

52 

456 

342 

53 

54S 

46 

566 

456 

^5 

634 

35 

345 

563    , 

36 

563 

372 

4,056 

4.797    421 

4,438 

See  P.  Ed.,  p.  77. 

1 26  FIRST  STEPS  AMOMC  nGURE& 


(«») 

(■3) 

(•4) 

(•S) 

(16) 

(■7) 

(18) 

% 

65^ 

♦S 

546 

544 

45 

S3 

566 

54 

354 

654 

6 

64 

626 

345 

66 

66s 

565 

55 

55 

356 

636 

36 

^36 

666 

63 

46 

465 

563 

45 

565 

3^5 

34 

3i 

SS6 

454 

56 

653 

456 

65 

56 

3,022  3,216  302  3.019  3,210  268  307 


(I) 

(») 

(3) 

(4) 

(5) 

{^) 

(7) 

354 

453 

24 

«5 

546 

36 

26 

45 

343 

45 

344 

365 

56 

S3 

533 

545 

34 

23^ 

653 

63 

66 

542 

434 

53 

544 

546 

-s 

32 

324 

455 

44 

355 

53 

66 

65 

453 

44 

35 

243 

466 

54 

66 

545 

532 

33 

425 

232 

46 

26 

343 

341 

42 

544 

665 

35 

63 

45^ 

553 

54 

43' 

536 

63 

66 

344 

434 

33 

554 

462 

56 

55 

533 

345 

21 

345 

355 

6 

43 

234 

25 

54 

423 

646 

63 

66 

543 

544 

35 

345 

355 

45 

35 

5,245  5,048     507  4,800  s,88o     614      662 
Sec  P.  E<L,  p.  7& 


FIRST  ^TEPS  AMONG  FIGURES. 


127 


The  answers  to  the  examples  are  at  the  end 
of  the  book.  They  are  placed  there  so  that  if 
the  teacher  wishes  any  pupil  of  the  class  to 
copy  the  examples  on  the  blackboard  for  him, 
it  may  be  done  without  the  pupil's  knowing 
what  the  answer  is.  If  the  teacher  prefers  to 
have  the  answers  with  the  examples  he  can 
copy  them  from  the  end  of  the  book. 

In  adding  long  columns  of  figures  it  is  well 
to  write  the  sum  of  each  column  56 

separately,  as  follows,  so  that  in  34 

adding  each  way  for  proof  the       120 
sum  of  each  column  may  be  seen       57 

at  a  glance.  

69396 

Show  the  pupils  that  when  9  is  added  to  any 
number  the  unit  figure  of  this  sum  will  be  one 
less  than  t.^e  unit  figure  of  the  number,  thus : 
9  +  37  is  46,  74  +  9  is  83,  &c. 

34  +  6?     75+9?     26  +  5.^ 


18  +  4? 
36  +  9? 
64  +  7? 
68  +  5? 

56  +  3? 


47+9? 
67+5? 
45+3? 
59  +  8? 
47  +  6? 


24+3 
48  +  8 
26-1-6 

74  +  4 
78  +  9 


47^8? 
85+6I 

29+4? 
87+9? 
85  ♦-7? 
69+5? 


128 


FIRST  STEPS  AMONG  FIGURES. 


74  +  8? 

3S+4? 

56  +  7? 

37+3? 

78  +  6? 

59  +  9? 

75  +  4? 

4S+81 

26  +  4? 

57  +  7? 

78  +  3? 

59  +  6? 

84  +  9? 

65+5? 

46  +  8? 

87+4? 

68  +  7  ? 

49  +  3? 

Give  much  practice  on  the  above  exercise ;  it 
will  be  of  great  use  to  the  pupil  in  all  additions. 

8.  463,075  X  465  =  ?  9-  640,753  X  306=  ? 


10.  560,423x204=  ? 
12.  574.863x37=? 
14.  680,574x78=? 


II.  67,052  X  234=? 
13-  867,534x56=? 
15-  475.806  X  406=  ? 


Before  taking  the  next  examples  give  the 
pupils  a  thorough  drill  in  division  with  re- 
mainder. 


Division  series  with  remainder.     (6  and  rev.) 


a 

b 

C 

23  103  25  8 
43266 

32  II  II  7 
'5423 

27   22    7    17 
4523 

d 

e 

f 

21  8  17  20 
6543 

5  33  17  13 
2652 

6  13  15  26 

4365 

( 

g 
15  4  9  39  12 
43265 
>ee  P.  Ed.,  p.  80. 

FIRST  STEPS  AMONG  FIGURES. 


129 


7's  (and  review)  with  remainder. 

a 

22  19  42  34 

3456 

b 

69  19  14  39 

7    3    4    5 

c 

28  62  17  39 

6734 

d 

e 

f 

32  23  53  13 
5673 

34  29  57  47 
4567 

II  31  21  52 
3456 

g 

51  2827  17 

7    3    4    '5 

h 

47  32  26  21 

6734 

i 

49  39  25 

5    6    7 

8's  (and  review)  with  remainder, 
a  I  b 

55  26  45  24      I     34  16  31  53 
8765      I       4387 


4'  69  13  31 
7834 

g 

26  28  63  32 

4387 


20  39  28  53 

3    4    5^ 

h 

49  33  61  39 
5678 


29  43   2  1    29 

6543 

f 

76  47  22  38 
8765 

i 
5831  14  23 
6443 


19  39  69 

5    6    7 


k 

44  26  17 

8    3    4 


I.   1,396,897-3=?  2.  1,821,287-7-5=? 

3.   2,144,698-4=?  4.  39,164,794^6=? 

5.  22,538,618-7-6=?  6.  27,6^2,523^6=? 
9 


130  FIRST  STEPS  AMONG  FIGURES. 

7.   10,763,483-^-7=?      8.  45.171  547-7=? 
9-  3i,752,20o-i-7=  ?    10.  508,019,899-^-8=? 
II.  34.856,549-^8=?    12.  61,476,243-^-8=? 

Unless  the  pupils  solve  the  foregoing  examples 
readily,  they  should  review  them  at  once. 


LONG  DIVISION. 

The  teacher  may  say  to  the  pupils  that  when 
the  divisor  is  a  large  number  the  method  of 
short  division  is  too  difficult,  illustrating  by  an 
example. 

Teach  the  pupils  that  the  first  step  in  solv- 
ing an  example  in  which  the  divisor  is  greater 
than  12,  is  to  place  a  comma  after  the  first 
figure  in  the  divisor  as  in  the  example, 
5,02)73245.  As  in  short  division  we  cannot 
divide  the  whole  of  a  large  dividend  at  once, 
so  we  cannot  in  long  division.  The  next  step 
is  to  find  how  much  of  the  dividend  we  will 
divide  at  first.  See  if  the  first  figure  of  the 
divisor  is  less  than  the  first  figure  of  dividend 
or  whether  it  is  greater.  In  the  example 
given  it  is  less,  (5  being  less  than  7).  Teach 
the  pupils  that  when  it  is  less  they  are  to 
count  as  many  figures   in   the  left  of  the  divi- 


FIRST  STEPS  AMONG  FIGURES.  I3I 

dend  as  there  are  figures  in  the  divisor  and 
place  a  comma  after  the  last  figure.  In  the 
given  example  there  are  three  figures  in  the 
divisor,  so  count  3  figures  in  the  left  of  the 
dividend  and  writing  a  comma  there,  the 
example  becomes  5,02)782,45. 

Require  the  pupils  to  take  these  two  steps, 
(and  no  more)  with  the  following  examples, 
first  placing  them  in  form  for  dividing  : 

763+5-^4321-         5738-^49- 
9458-^875.     875988^58841.     748567^-2145. 
57881-468.       76854-68.     8768456 -i- 25864. 

Teach  the  pupils  that  if  the  first  figure  of 
the  divisor  is  greater  than  the  first  figure  of  the 
dividend  we  count  one  more  figure  in  the  divi- 
dend than  there  are  figures  in  the  divisor,  in 
order  that  the  part  we  take  may  be  large  enough 
to  contain  the  divisor.  *  In  the  example  687531 
—  7342  the  first  figure  of  the  divisor,  7,  being 
greater  than  the  first  figure  of  the  dividend,  6, 
we  count  one  more  figure  in  the  dividend  than 
the  4  figures  there  are  in  the  divisor  and  the 
example  with  these  two  steps  taken  becomes 
7,342)68753,1.  Require  the  pupils  to  take 
these  2  steps  with  the  following  examples  and 
•  See  P.  Ed.,  p.  98. 


132  FIRST  STEPS  AMONG  FIGURES. 

more  if  they  are  needed  to   make  the  entire 
class  familiar  with  these  steps ; 

57281456-71235. 

423214-^5347. 

342165-^2543. 

6847-71. 

5875643^643. 

68475032-^934674. 

47325684^3145. 

The  next  step  is  to  count  the  number  of 
figures  at  the  right  of  the  comma  in  the  divisor 
and  count  the  same  number  of  figures  at  the 
left  of  the  comma  in  the  dividend,  and  place  a 
comma  before  the  one  counted  which  is  farthest 
to  the  left:  thus,  in  the  example  34276^404, 
the  first  step  is  4,03)34276  ;  the  second, 
4,03)3427,6  ;  the  third,  4,03)34,27,6. 

Require  the  pupils  to  take  these  steps  (and 
no  more)  with  the  following  examples : 

674532-^5342. 

75694857-^845321. 

546327-^643. 

345367471-^75382. 

47346-T-23. 

4326472-^6351. 


FIRST  STEPS  AMONG  FIGURES.  I33 

8345376-H702. 

56341-5-68. 

763542^8547. 

57345^493- 
In  the  example  631663-7-201,  which  after  the 
three  steps  is  2,01)6,31,663,  the  next  step  is, 
-see  how  many  times  the  number  at  the  left  of 
the  comma  in  the  divisor  is  contained  in  the 
Tiumber  at  the  left  of  the  first  comma  in  the 
"dividend.  2  is  contained  in  6  three  times. 
The  example  becomes  2,01)631,663(3.  Next 
multiply  the  divisor  by  this  quotient  figure, 
placing  the  first  figure  of  the  product  under  the 
figure  before  the  last  comma,  thus : 

2,01)6,31,663(3 
603 

Next  step  see  if  you  can  subtract.  (Teach 
the  pupils  to  look  at  the  left  hand  of  the  num- 
ibers  to  see  whether  they  can  subtract.  If  the 
pupils  ask  what  is  to  be  done  when  you  cannot 
-subtract,  tell  them  you  will  show  them  in  the 
first  case  in  which  they  cannot  subtract,  which 
will  not  occur  in  the  examples  given  for  some 
time.)  Next  subtract.  (Show  the  pupils  that 
the  remainder  should  be  less  than  the  divisor.) 


134  FIRST  STEPS  AMONG  FIGURES. 

Next  see  that  the  remainder  is  smaller  than  the 
divisor. 

(Do  not  show  the  pupils  what  to  do  when  the 
remainder  is  larger  than  the  divisor  until  a  case 
occurs  in  their  work.)  Write  the  next  figure 
of  the  dividend  at  the  right  of  the  remainder. 

2,01)6,31,663(3 
603 

286 

Next  step  count  as  many  figures  from  the 
right  of  the  partial  dividend  as  there  are  at  the 
right  of  the  comma  in  the  divisor  and  the  ex- 
ample becomes 

2,01)6,31,663(3 
603 

2,86 

Divide  as  at  first  and  so  continue  the  opera- 
tion. 

The  steps  are : 

I  St.  Write  the  divisor  and  dividend  in  the 
proper  form. 

2d.  Point  off  in  the  divisor. 

3d.  Place  the  right  hand  comma  in  the  divi- 
dend. 


FIRST  STEPS  AMONG  FIGURES.  135 

^^^  ^_^_ ^ ■» 

4th.  Place  the  left  hand  comma  in  the  div- 
idend. 

5th.   Divide. 

6th.  Multiply. 

7th.  See  if  you  can  subtract 

8th.  Subtract. 

9th.  See  that  the  remainder  is  less  than  the 
divisor. 

10th.  Write  the  next  figure  of  the  dividend. 

nth.  Point  off. 

Repeat  steps  5,  6,  7,  8,  9,  10  and  11  until  the 
example  is  solved. 

The  teacher  should  solve  the  following  three 
examples  7///M '  the  pupils,  before  any  are 
given  them  to  solve  alone.  The  teacher  taking 
the  crayon,  the  pupils  will  tell  what  is  to  be 
done,  one  pupil  describing  the  first  step, 
another  the  second  and  so  on,  or  better  yet, 
one  0/  them  take  one  of  the  steps  then  another 
pupil  take  another  and  so  on.  First  solve 
twice  the  example  given  in  the  foregoing 
illustration. 

•  For  method : 

•  The  pupils  should  erase  the  commas  which  divide 
the  number  into  periods  before  pointing  off,  that  there 
be  no  confusion.  The  commas  for  the  operation  of  di- 
viding may  be  placed  above  the  number  instead  of 
beneath  it,  if  preferred. 


136  FIRST  STEPS  AMONG   FIGURES. 

1.  5,450,204-7-403=? 

2.  162,479,845-3042=1 

3.  20,913.844^604=? 

Do  not  give  the  pupils  more  than  one  or  two 
examples  each  day  until  you  are  sure  they 
understand  the  method. 

4-  4.920,352-7-2,023=?     Ans.  2,432*'^ 

5.  16,312,418^5,013=?     Ans.  3,254"* 

6.  273,785,577-^60,345=? 

7.  26,308,025-7-4,063=? 

8.  189,771,597-^-50,364=? 

9.  524,601,734-^6,047=  ? 

10.  39 1,838,602  -f-80,675  =  ^ 

11.  619.307,367^80,597=? 

12.  278,696,736^6.075=? 

13-  45 +  363 +456 +  542+6 -f- 356 -f- 663  + 

454  +  32+46  +  553  +  636+45=  ? 

14-  465+564  +  32+646  +  553+465+566-f- 
632-^665+356-^43+655+6=  ? 

15.  4264-563  +  365+634  +  5454-643+356 
266  +  633  +  56  +  445+54  +  63=  't 
I..  A  fox  caught  5  geese  which  were   1-3  of 
the    farmer's   flock ;   how   many  geese  in  the 
flock? 

2.  A   hen  had  15  chickens  ;  a  cat  caught  4 
of  them  and  a  hawk  3.     How  many  were  left  ? 
See  P.  Ed.,  p.  82. 


FIRST  STEPS  AMONG  FIGURES.  137 

3.  Arthur  was  paid  14  cents  for  doing  an 
errand  ;  he  lost  5  cents  and  his  sister  gave  him 
7.     How  many  cents  had  he  then  ? 

4.  A  squirrel  carried  4  nuts  home  one  day  J 
5  the  next  day  and  on  the  third  enough  to 
make  his  number  16.  How  many  did  he 
carry  home  the  third  day  ? 

5.  James  has  18  apples  to  divide  equally 
among  3  boys  ;  how  many  shall  he  give  to 
each? 

6.  How  many  yards  of  tape  at  2  cents  a 
yard  can  I  buy  for  15  cents  and  have  i  cent 
left? 

7.  Fred  gave  5  cents  for  an  orange,  18  cents 
for  figs  and  4  cents  for  a  lead  pencil.  How 
many  cents  did  he  spend  ? 

8.  In  an  orchard  the  trees  were  set  16  in  a 
row ;  7  in  each  row  died.  How  many  living 
trees  in  each  row  ? 

9.  A  boy  sold  a  pair  of  doves  for  25  cents 
and  bought  as  many  marbles  at  3  cents  apiece 
as  he  could  for  the  money.  How  many 
marbles  did  he  get  and  how  many  cents  left? 

10.  A  boy  earned  9  cents  one  day  and  12 
cents  the  next.  How  much  more  did  he  earn 
the  second  day  than  the  tirst .' 


138  FIRST  STEPS  AMONG  FIGURES. 

11.  Frank  had  16  rabbits  ;  he  sold  3  to  one 
boy  and  4  to  another.  How  many  did  he 
keep? 

12.  How  long  will  it  take  a  miller  to  grind 
42  bushels  of  grain  if  he  grinds  6  bushels  in 
an  hour  ? 

13.  Willie  bought  2  pass-books  at  5  cents 
each  ;  a  lead  pencil  for  6  cents,  and  3  oransjes 
at  4  cents  a  piece.     What   did  he  pay  for  all  > 

14.  Willie  keeps  rabbits  to  sell ;  he  has  20* 
and  has  4  little  houses  for  them.  How  many 
does  he  keep  in  each  house  ? 

15.  If  4  bags  contain  8  bushels  of  grain,  how- 
many  bushels  will  9  bags  hold  ? 

16.  If  5  cords  of  wood  cost  130,  what  will  3; 
cords  cost  ? 

17.  If  it  cost  15  cents  to  ride  5  miles  on  the 
cars,  how  much  will  it  cost  to  ride  7  miles  ? 

18.  How  many  bushels  of  oats  will  3  horses 
eat  in  a  week,  if  6  horses  eat  42  bushels  in  a 
week  ? 

19.  If  6  brooms  cost  18  shillings,  what  will 
5  brooms  cost  ? 

20.  If  a  barrel  of  flour  will  last  2  men  6 
months,  how  long  will  it  last  i  man  ? 

21.  If  2  men  consume  6  barrels  of  flour  in  a 

See  P.  Ed.,  p.  86. 


m 


FIRST  STEPS  AMONG  FIGURES.  1 39 

certain  time,  how  much  will  i  man  consume  in 
the  same  time  ? 

22.  If  4  horses  eat  12  bushels  of  oats  in  5 
days,  how  many  bushels  will  i  horse  eat  in  the 
same  time  ? 

23.  If  3  teams  will  plow  a  certain  field  in  6 
days,  in  how  many  days  will  i  team  plow  it  ? 

24.  If  4  men  can  dig  a  certain  ditch  in  & 
days,  how  long  will  it  take  2  men  to  dig  it  ? 

25.  If  3  men  cut  6  cords  of  wood  in  a  day, 
how  much  will  9  men  cut  in  a  day  ? 

26.  If  it  take  3  men  6  days  to  cut  a  pile  of 
wood,  how  long  will  it  take  9  men  ? 

27.  A  man  lost  $6  by  selling  a  cow  for  $37  -^ 
what  did  the  cow  cost  him  ? 

28.  A  boy  sold  4  pencils  al  2  cents  each,  and 
3  marbles  at  3  cents  each  ;  how  much  money 
should  he  receive  ? 

29.  What  is  the  wheat  in  7  bags  worth  at  $2 
a  bushel,  if  there  are  2  bushels  in  each  bag  ? 

Read  the  following  : 

1.  70000580030. 

2.  68000C50000. 

3.  680507415371. 

4.  756847597547. 

5.  76500000068. 

6.  67459800000. 


140  FIRST  STEPS  AMONG  FIGURES. 

Teach  pupils  to  write  billion. 

1.  Write  in  Arabic,  seventy  million  three 
thousand  forty. 

2.  Write  in   Arabic,  five  billion  four  mil- 
lion nineteen. 

3.  Write  in  Arabic,  ten  billion  three  hun- 
dred million  fifty  thousand. 

4.  Write   in    Arabic,   fifteen    billion    nine 
thousand. 

5.  Write  in  Arabic,  two  hundred  billion  ten 
million  twenty. 

6.  Write   in    Arabic,    forty  million  twenty 
thousand. 

7.  Write  in  Arabic,  nine  hundred  forty  bil- 
lion one  hundred  six  thousand  five  hundred. 

8.  Write  in  Arabic,  sixteen  billion  sixteen. 

9.  Write  in  Arabic,  five  billion  forty  mil- 
lion. 

10.  Write  in   Arabic,  nine  hundred  billion 
nine. 

11.  Write    in  Arabic,  eight    billion    ninety 
thousand  four. 

Teach  Roman  to  1880. 

12.  Write   in  Roman,  one    thousand    three 
hundred  forty-one. 

13.  Write  in  Roman,  nine  hundred  seventy- 
six. 

See  P.  Ed.,  p.  88. 


FIRST  STEPS  AMONG  FIGURES.  I4I 

14.  Write  in  Arabic  and  in  words  MDCCL- 
XXV. 

15.  Write  in  Arabic,  ten  billion. 

16.  Write  in  Roman,  1876. 

1 7.  Write  in  Roman,  nine  hundred  forty-nine* 

18.  Write  in  words  761308260017. 

20.  Write  6  units  of  the  8ih  order,  8  of  the- 
5lh,  3  of  the  3d  and  5  of  the  ist. 

21.  Express  the  following  number  by  naming 
the  units  and  their  order,  beginning  at  the  left  '^ 
70900048010. 

For  practice  in  subtraction,  to  be  given  to  the 
pupils  orally  and  recited  as  the  series  have 
been. 

49  (To  be  read,  45  from  49?)       \\    \\    5| 

45     69     52     38     62     35     66     19     51     22 

37  66     45     34     55     29     65     14     43     19 

45     82     56     32     59     36     51     85     31     56^ 

38  66     46     28     53     29     46     66     25     54 

38     73     18     30     72     46     47     63     30     57 
33     67     16     25     64     43     38     59     23     54 

43     50     32     64     48     61     67     48     54     39 
25     48     27     56     45     54     63     39     48     37 


142  FIRST  STEPS  AMONG  FIGURES. 

31       70      34      50       24      51       67       65       84      56 
26      64      25       46       17      48       59       63       79       48 

30     93 
27     86 

For  rapid  solving.     (To  be  read.) 

I.  3  +  5  +  9  +  7+8-4-4-6x9  +  9  +  54-7-1- 
6-3-+7X  9-8-8-3-7 +8 +  5-i-3 
X8  +  9  — 4— 5  — 7=?  Ans.  41. 
J.  65-7-5-5-3^9x7-9-7-4  +  5 
X9+9— 8  +  7x9+  8—6  —  2-6x7 
—5-5+4x3=?     Ans.  64. 

3.  6I-5-9-5-7  +  9  +  I+-9  +  7+-3X9 

+  6— 7-8+3X  8-9-7-9-5-7 
X4  +  7+4-'6=?     Ans.  29. 

4.  9  x8— 6  — 5  — 7-^9  X4  — 8  X6  +  6+-4X 

7  +  3+9x6  — 2-^4x6  — 7-4-7x8  +  2 
-+6x4  +  4-8=.'     Ans.  4. 

5.  4x3  +  8-5-4x7-3^8x5  +  4-6x7  + 

8-7-9x6  +  6  +  5x3  +  2+4  +  2+6  +  8 

-+5  +9=  ?     Ans.  17. 
^.  5X7  +  7-^6x4  +  7^7x6-4-5+3X 

4  +  4-^8x5+6  +  6-^4x6—3-^5x3 

+  8-4-5  +  9=  ?     Ans.  i6. 
7-  5x7-3^8x6  +  6+5x8—3-9x6  — 

6-4-4x7—6-8^7x5+6-1-4-5+3 
X7  — 2H-6=.'     Ans.  9. 


FIRST  STEPS  AMONG  FIGURES.  I43 

8.  4x6  +  8-^8x9  —  4^-8x9—94-3-1-7-5- 

4=  ?     Ans.  4. 
9-  6X3  +  74-4  +  7-5-9X4  +  5-^3><6-6-7- 
6  X  8  +  6-T-9  X  7— 8— 6-r-4  X  3 -t-3-i-8 
X9--2-^5  x6  — 7  — 7=  ?     Ans.  16. 
10.  4-f-7-}-6  +  9  +  8-7-f-3-f-8-h8-r-5x6- 
6-8-T-4X8— 7-6  — 7^3  X9  +  9-J-5 
X9  — 7— 8=  ?     Ans.  66. 
Problems  for  the  slite  involving  Addition, 
Subtraction  and  Multiplication  : 

1.  There  are  320  rods  in  a  mile  ;  how  many 
rods  in  79  miles  ? 

2.  *  William  has  75  cents  and  Charbs  has 
'68  cents  more  than  William  ;  how  many  cents 
have  both  boys  ? 

3.  David  had  123  cents  ;  he  spent  47  cents 
for  a  ball  and  39  cents  for  marbles.  How 
many  cents  had  he  left? 

4.  The  larger  of  two  numbers  is  916  and 
their  difference  is  43  ;  what  is  the  less  number  ? 

5.  If  the  drive  wheel  of  a  locomotive  turn 
around  352  times  in  going  i  mile,  how  many 
times  will  it  turn  around  in  going  from  Canan- 
daigua  to  Rochester,  the  distance  being  29 
smiles? 

•  In  the  different  steps  of  such  an  example  it  is  im- 
.portant  as  a  help  to  mark  each  result. 
See  P.  Ed.,  p.  90. 


144  FIRST  STEPS  AMONG  FIGURES. 

6.  A  farmer  having  239  sheep,  sold  99  of 
them  and  then  bought  113  ;  how  many  had  he 
then? 

7.  There  are  5280  feet  in  a  mile  j  how  many 
feet  in  357  miles? 

1.  374  +  645+57  +  767+436  +  543+675  + 
7  +  454  +  765  +  577  +  456=? 

2.  45  +  776  +  567  +  457  +  734  +  475  +  67  + 
754  +  5  +  676  +  547  +  375=  ? 

3-  6  +  347  +  75  +  657  +  746+773  +  46  +  457 

+  347  +  675  +  766  +  577=? 

4-  576+745  +  457  +  674  +  77  +  556  +  473  + 
765  +  7  +  657  +  564+705  +  76=1 

5-  546  +  375  +  657  +  74  +  765  +  257  +  6  +  75 
+  743  +  656  +  777  +  467  +  762=? 

6.  647  +  756  +  475+367  +  636  +  753  +  77  + 
345  +  676  +  45  +  576  +  767  +  654  +  365=^ 
7-  6,314,532-521,987=? 

8.  463,524-39-643=  ? 

9.  653,425-64,287=? 

10.  475*067-36,543=? 

11.  688,045-95,387=? 

12.  4,760,352-376,534=? 

13.  4,630,024—921,045=? 

14.  34,000,435  —  2,700,518? 

See  P.  Ed.,  p.  92. 


FIRST  StEPS  AMONG  FIGURES. 


MS 


15-  4,500^375-760,187=  ? 

16.  354,000,253  —  272,102,437=1 

17.  530,024-543,052=? 


1.  89,756x96=? 

2.  364.758x356=? 

3.  638,497  x68=? 

4.  498.675x97=? 

5.  60,847  X  708=  f 

6.  796,805x705=? 

7.  807,009  x6o8=  ? 

8.  6,859x748=  ? 

9.  94,786  X  7,968=  ? 
10.  603,405  —  612,134=? 
II-  6,537,065-743,987=? 


Division  series  with  remainder. 


9*s  and  review. 

a 

b 

32   15  68  39  61 
54987 

33  49  26  46  77 
65489 

c 

d 

19  37  21  46  78 
45678 

52  53  27  39  61 
96549 

e 

f 

28  54  29-34  61 
87678 

35  58  41  70  43 
96789 

146 


FIRST  STEPS  AMONG  FIGURES. 


g 

29  18  40  68  43 

45675 


21  88  54  25 

4987 


47  23  34 
6     5     4 

1.  33.256,023-7-7=? 

2.  39,044,761-^6=1 

3-  34,352.839-^5='^ 

4.  27,832,074-6=? 

5.  445.941,095-^7=1 

6.  27,516,279-5-4=  ? 

7.  449,181,483-^6=? 

«.  487>959'992-^7=? 

9.  28,556,208-6=? 

10.  15,228,723^4,058=? 

11.  251,776,292-^70,486=? 

12.  386,124 633-^6o,578=? 

13.  41,847,116^9,048=? 

14.  52o,6I3,47I-^ 70,697=? 

Teach  the  pupils  that  when  any  partial  div- 
idend (after  writing  the  next  figure  of  the  div- 
idend at  the  right  of  the  remainder)  is  less  than 
the  divisor,  they  must  write  a  cipher  in  the  quo- 
tient, just  as  they  do  in  short  division.  Next 
See  P.  Ed.,  p.  96. 


FIRST  STEPS  AMONG  FIGURES.  I47 

erase  the  comma  made  in  pointing  off  the  par- 
tial dividend,  write  the  next  figure  of  the  divi- 
dend at  the  right  of  the  partial  dividend,  point 
off  and  proceed  as  before. 

15.  276,265,200^4,036=  ? 

16.  42,865,597^-6,075='? 
17-  397,706,673-^6,053=? 

18.  44,i53,4i9,6i9-=-7o,386=f 

19.  15,086,456-7-4,023=  ? 

20.  4,437,512,234-^-60,289=? 

In  the  preceding  examples  the  second  figure 
from  the  left  of  the  divisor  has  in  each  case 
been  a  cipher,  and  the  examples  have  been 
so  constructed  that  the  divisor  is  con- 
tained as  many  times  in  each  partial  dividend 
as  it  appears  to  be.  A  new  difficulty  will  arise 
in  the  following  examples  since  the  second 
figure  of  the  divisor  is  a  significant  figure. 

Show  the  pupils  that  when  the  second  figure 
has  value,  the  divisor  is  often  not  contained  in 
the  partial  dividend  as  many  times  as  it  ap- 
pears to  be,  since  in  multiplying  the  divisor  by 
the  quotient  figure  there  will  usually  be  some- 
thing to  add  to  the  product  of  multiplying  the 
first  figure,  coming  from  the  product  of  multi- 
plying the  second.    Teach  the  pupilsto  observe 


148  FIRST  STEPS  AMONG  FIGURES. 

how  much  there  will   be  to  add  to  this  first 
product  and  to  allow  for  it. 

Teach  them  also  that  when  any  product  is 
greater  than  the  partial  dividend,  it  shows  that 
the  quotient  figure  which  gave  that  product  is 
too  large,  and  that  the  partial  product  and  that 
quotient  figure  must  be  erased  and  a  smaller 
quotient  figure  used. 

Use  the  following  examples  in  illustration  to 
the  class,  or  at  least  as  many  examples  as  will 
make  the  matter  clear  to  the  class : 
267,142  —  352=  } 
2,659,478^461=? 

Use  too  large  a  quotient  figure  in  some  in- 
stances so  as  to  show  the  pupils  what  to  do 
when  they  use  one  that  is  too  large.  Teach 
them  to  use  much  care  in  finding  the  quotient 
figure  and  so  save  themselves  much  work. 
Show  that  the  divisor  never  should  be  con- 
tained in  the  partial  dividend  10  times. 

1.  4.367,695^673=? 

2.  220,3964-254=  ? 

3.  307,627^354=? 

4.  4,925.151-^-694=? 

5.  6,310.318-^781=? 

6.  3,280,381-^482=? 

See  P.  Ed.,  p.  98. 


FIRST  STEPS  AMONG  FIGURES.  149 

7.  239,294,268-^3,642=? 

8.  334/;90,637  4-574=  ? 

For  rapid  solving.    (To  be  read  and  to  be 
answered  without  use  of  slate.) 

1.  6x6  —  8-1-4x6  +  8  —  2-7-6x4+8-7-5x9 

—9+9x6  +  8  +  9=?  Ans.  59. 

2.  9x6  — 5-+7  x6  —  6-r-4X  7  +  5  +  4-9+9 

+  6  +  8-4-7-3=?  Ans.  9. 

3.  5X8-6-7+-3X5  +  7+6  +  5+9X6+3 

^5x3  +  9  —  7=  ?  Ans.  29. 

4.  8x7-9-5-+6x3-3  +  3X7  +  3  +  5x6 

—  6  — 6-j-6x4=?  Ans.  28. 

5.  6x8-6-7-^-5x8  +  4  +  3  +  7x4  +  7+5 

-8  +  7  +  5+9=  ?  Ans.  27. 

6.  4x9  +  8  +  4^-6x4—5+3x6—8-4+6 

x5  +  8  +  4=.?  Ans.  47. 

7.  7  x6  — 5— 9-+4X8— 8— 6  — 2— 8  X5 +^ 

+  7-8=?  Ans.  33. 

8.  6x4  +  8-^8x9  +  6  +  7x8  +  1-7-7x9-4 

-3-+7  +  9=^  Ans.  17. 

9.  9x8  — 6—9  — 1-:-7  x5  +  5-^9  X  6  +  8+4 

-^6x8  +  7 -^9x4  +  6  +  9  +  5+8  X  6 

+  7  +  2-7-9x8-8  —  5+9=?  Ans.  3. 

10.  7  x6— 6  +  9-+5  X  6— 8— 9+5-7-7  x8— 6 

-5 -5 +-8x9  +  9-7  +  6+9+3+8 


150  FIRST  STEPS  AMONG  FIGURES. 

X  4  +  9  +8-^9x7  +  8  +  5-^8x7=? 

Ans.  42. 
II.  9x6  —  8  —  7  +  6+9x6  —  6+3x7  +  8  —  7 

+  6^7  X  4  +  9  +  8  — 7— 4+-6x8  — 7 

-6-7  +  4  +  7+8+  8  +4  x7+8=? 

Ans.  64. 
la.  8x8-+ 6  +  3  — 7  —  8— 4+-6X7-8— 6-1-7 

X8-8+6X9  +  3— 8— 4+-9+8  +  9  + 

7  +  6  +  9  —  7  +  5  +  8  —  7=1      Ans.  45. 

13.  6x8  +  3-7-8  +  9x7  +  9  +  8  +  5x6  + 

2  +  7x9  —  7—9  —8-8x7— 8  — 7-+3 
+  7  +  8  +  7=?     Ans.  31. 

14.  7x9-8-7-4-8x9+8-5-6-4—5  + 

6X8  +  9  +  7+-9  +  9+  9  +8  +  2-^4x7 
—  7-^8x5+8  +  9  +  8=?     Ans.  60. 

1.  How  far  will  a  boy  walk  in  7  days, 
walking  9  miles  each  day  ? 

2.  How  far  will  a  boy  walk  in  2  days^ 
walking  9  miles  the  first  day  and  7  miles  the 
second  day? 

3.  In  how  many  days  will  a  man  earn  48 
shillings,  at  8  shillings  a  day  ? 

4.  A  boy  earned  45  cents  Monday,  and  53 
cents  Tuesday ;  how  much  more  did  he  earn 
Tuesday  than  Monday  1 

See  P.  Ed.,  p.  99. 


FIRST  STEPS  AMONG  FIGURES.  151 

5.  If  4  peaches  cost  8  cents,  what  cost  9 
peaches  ? 

6.  If  a  merchant  sells  9  spools  of  thread  in 
5  hours,  at  that  rate  how  many  would  he  sell 
in  1  hour  ? 

7.  If  3  girls  can  make  6  aprons  in  a  day, 
how  many  can  one  girl  make  in  a  day  ? 

8.  If  2  girls  can  do  a  piece  of  work  in  8 
days,    in  how  many  days  can  i  girl  do  it? 

9.  A  boy  bought  9  marbles  and  lost  all  of 
them  but  3  ;  how  many  did  he  lose  ? 

10.  A  farmer  bought  a  pig  for  $6  and  sold  it 
for  $9  ;  how  much  did  he  gain  ? 

11.  There  were  8  cows  in  a  field  and  6  more 
were  put  in  ;  how  many  were  in  the  field  then? 

1 2.  There  are  1 7  girls  in  a  class  and  9  boys ; 
how  many  pupils  in  the  class  ? 

13.  There  are  16  caps  in  the  entry  and  7 
bonnets  ;  how  many  more  caps  than  bonnets  in 
the  entry  ? 

14.  If  John  is  well  how  many  days  should 
he  come  to  school  in  4  weeks  ? 

15.  A  man  may  rightly  work  how  many  days 
in  3  weeks  ? 

16.  James  has  5  cents  and  his  sister  has  2 
cents  less  than  twice  as  many ;  how  many  have 
both? 


152  FIRST  STEPS  AMONG  FIGURES. 

Examples  for  slate  in  Addition,  Subtraction, 
Multiplication  and  Division : 

1.  A  has  9  fields,  containing  in  all  197  acres  j 
B  has  13  fields,  containing  239  acres  ;  C  has 
17  fields,  containing  298  acres;  D  has  6  fields, 
containing  85  acres  ;  how  many  fields  and 
how  many  acres  have  all  ? 

2.  How  many  horses  at  $185  each  can  be 
bought  for  25  cows  at  $37  each  ? 

3.  If  69  acres  of  land  cost  $6,486  what  will 
207  acres  cost  ? 

4.  I  borrowed  of  Mr.  Rawson  at  one  time 
$697,  at  another  $1,748,  and  at  another  $456; 
I  paid  him  $975 ;  how  much  do  I  still  owe 
him? 

5.  What  is  the  sum  of  eighteen  thousand 
three,  nine  million  twenty  thousand,  eight  hun 
dred  six,  seven  thousand  sixty,  95  thousand 
seven  hundred,  twenty-one  million  five  hundred 
seventy-six,  and  ten  million  ten  ? 

6.  Henr)''s  kite  was  up  in  the  air  375  feet,  it 
then  fell  98  feet  and  then  rose  268  feet ;  how 
high  was  it  then  ? 

7.  Three  men  bought  a  hotel  for  $25,800; 
the  first  paid  $6,790,  the  second  twice  as  much, 

See  P.  Ed.,  p.  102. 


FIRST  STEPS  AMONG  FIGURES.  153 

and   the  third  the  remainder  ;    how  much  did 
the  third  pay  ? 

8.  The  earnings  of  a  father  and  his  3  sons 
for  a  year  amount  to  $2175  ;  their  expenses  are 
^957  >  'f  ^^^  balance  is  divided  equally,  how 
much  will  each  have  ? 

9.  If  four  dresses  of  15  yards  each  are  cut 
from  78  yards  of  calico,  how  many  yards  will 
be  left  ? 

1.  7+46+74  +  35+47  +  73+644-574-75 
+  6  +  77  +  45+63  +  57  +  46  +  75=? 

a.  376+455  +757+463+375  +  747+654 
+  37  +  576  +  346  +  775  +  7-^464  +  647 
+  356  +  565=? 

3.  475  +  647  +  756  +  765+437  ^674  +  575 

+  756  +647+  567+456  +  743  +357 
+  556  +  463  +  756=  ? 

4.  76  +  35  +  8  +  47+  85  +37  +  56  +  87  +  75 

+  84  +  68=  ? 

S-  58+765  -485+  678  +537+6  +  753  + 
488  +  846  +  587  +  758=? 

6.  678  +  845  +  784  +  326  +  487+856  +  678 

+  588  +  865+478  +  756=1 

7.  7  +  58  +  84  +  56  +  78  +  64  +  87+6  +  58  + 

75+86  +  68  +  75=? 


154  FIRST  STEPS  AMONG  FIGURES. 

8.  648 -I- 785  4-  874  +688  +  576  +  845+68$ 

+  786  +  75  +  847+687+58  +  6=? 

9-  758  +  875  +  684  +  768  +  475  +  886  +  744 
+  358  +652+  887+  546  +  785+64$ 

10.  8  +  57  +  68  +  84  +  75  +  7  +  58  +  76  +  88  + 

47  +  63  +  78  +  86  +  55+67+88=  ? 

11.  368+475  +  638  +  857  +  583+646  +  87$ 

57  +  645  -^  768  +  582  +  7  +  676  +  848 
587  +  766=? 

12.  67  +  788  +  856  +  475  +  687+878  +  564  + 

87  +  656  +  478  +  880  +  567  +  375  +  688 
856  +  785=? 

13-  7,963^034- 546,573=? 

14.  758,600341-79,423,275=1 

15.  8,460,075—987,286=  ? 

16.  658,000,468-35,030,273=? 

17-  43»750'078— 44.345,621=  ?    Impossible 
18.  6,475,000,374—293,030,596=1 

1.  97,806x59=  ? 

2.  97,865x896=? 

3.  96,897x6,978=? 
4-  5*740x7,500=? 

5.  746,800  X  9,000=  ? 

6.  470,900  X  70,580=  ? 

See  P.  Ed.,  p.  iia 


FIRST  STEPS  AMONG  FIGURES.  1 55 


7.  869,070  X  670,900=  ? 

8.  790,600x806,700=? 

9.  62,8o2,889-=-9=? 

10.  71,262,955-8=? 

11.  538,908,792-4-7=? 

12.  376.57i,o86-j-7=1 

13-  339,253.657-^9=? 

14.  324,763,528^7=? 

15.  446,2i7,i69H-9=  ? 

16.  54,284,406-7-8,047=  1 
17-  25,534,849-^7,I97=? 

18.  45,126,612-4-914=? 

19.  4,368,565-4-9,168  ? 
20.63,008,141-7-5,274=? 

21.  471230,943-^79=? 

22.  9,290,055.741-^4.869=1 
23-  331279,851-48,600? 

24.  68,643,216^87,000? 

25.  76,845,678-7-100=  ? 

26.  3.921,534,261-7-486,000=1 

27.  60,064,175-^8,000=? 

28.  90,700  X  50,700=  ? 

29.  Subtract  3  billion  6  thousand  750  from 
45  billion  I  million  seven  hundred  sixty-three 
thousand  4  hundred. 

30.  284,553,437^3,790=? 

See  P.  Ed.,  p.  116. 


'56 


FIRST  STEPS  AMONG  FIGURES. 


31.  4,167,300,326-5-4,790,000=? 

32.  5,074,000X68,070=? 

33-  3-424  330.02  » -^  497iOoo=  ? 

34.  2,468,576,216-^10,000=? 


12's  (and  review.) 
For  addition  and  multiplication, 
b 
7 


9  12     8  10 
8    9  10  II 

d 

3  12     8  12 

12  II   12     4 

g 

12     5   12 

3/26 


12 

7 
9 


II   9 
8  9 


e 
II     9   12 
10  II    12 


II 


9  12  8   10     7 
12     8  9   10   II    12 


8   10  7  II 
1 1    12  8     9 

I' 
8   10     7   II 
8     9  10  II 

i 

6   12     4 

12     5   12 


For  subtraction. 

a 
22  19  22  18  19  19 
12  II  10  9  8  12 

c 

16  21  23  19  20  15 

9  10  II  10  9  8 


21 

18 

2t 

17 

20 

18 

II 

10 

9 

d 

8 

12 

8 

20 

24 

16 

19 

17 

22 

II 

12 

8 

9 

10 

II 

21  20  17  20  23  18 
12  8  9  10  12  II 

See  P.  Ed.,  p.  140. 


FIRST  STEPS  AMONG  FIGURES. 


^57 


For  division. 

a 
72  121  108  96  72  100 
12  u   12  8  9  10 

c 
120  108  72  132  90  99 


32  77  80  110  84  4g 

12  1 1  10   1 1  12  12 

d 

56  88  120  81  96  80 

9  8  1 1  10  9   811   10  9  12  S 


e 
60  36 
12  12 


f 

63  no  99  144  64  90 

9  10  II   12  8  9 


I.  799,896  X  12=  ? 

2-  6,347,435  X  12=  ? 

3.  7,968,473  X  11=  ? 

4.  3,546,247  X  I2=? 

5.  989,769  X  12=/ 

6.  799.958  X  12=  ? 

7.  8,989,978  X  12=  ? 


12's  (and  review.) 
Division  with  remainders. 


127  76  94  70  150  108 

11  10    9     8     12     II 

c 
116  102  78  105  86  139 

12  8  9  10  II  12 


115  71  96  137  106  85 
10  9  10  II  12  8 

d 

124  88  94  95  117  89 

10  9  8  12  II  10 


See  P.  Ed.,  p.  117. 


«58 


FIRST  STEPS  AMONG  FIGURES. 


IIS  78  96  130  63  107 
9    8  II     12     8      9 


For  more  practice. 
For  division. 


^6  152  40  68  4- 

II  12  6  7  8 

c 

118  75  114  61  S^ 
12  II   987 


69  54  102  89  45 
67876 

g 

76  93  70  139  81 

689  II  12 


78  130  94  62  34 
9  1 1  12  7  6 


52  55  88  64  107 
6  8  9  II   12 

f 

142  86  52  78  48 

12  II  9  8  7 

h 

57  41  70  107 

6789 


8.  9,587,888,171 -fia=»t 

9.  95,621,647^12='? 

10.  813,764.564-4-11=? 

11.  83,645,840-12=1 

See  p.  Ed.,  p.  118. 


FIRST  STEPS  AMONG  FIGURES.  I59 

For  rapid  solving.     (To  be  read  and  to  be 
answered  without  the  use  of  slate.) 

I.  17+4^3x9  +  8  +  7-1-6-^12x9—8—6 
+  7-^7x6  +  9  +  6-^-9x11+9+8  +  7 
+  7-7-9  X  10-^-3  +  9+94-2  +  7-7-2  =  ? 
Ans.  18. 

a.  144-^2-7-2-^-9  x8x  2  — i-i-7  X  i2-i-3  + 
6-^6  X  12  X2+-3+-2  X3+9  +  9  +  3-r- 
2  +  6  +  7  X  4-7- 10  x8  -5-4x3-7-8=  ? 
Ans.  9. 

3.  9Xi2-2-^3X  2—44-2  X3  4-  12  +  7  X 

12-7-3  X  2-^4+  124-2x3  +  6-^3x2 
+  74-5x8-^3x24-3x44-8  +  9+8 
+  9x3  +  64-9=?     Ans.  12. 

4.  19x2  +  4  +  7x12-7-2x34-12x114-3 

X44-12  X  8  +  84-2 +34-3  X2+8-T-6 
XI2-^3X2  — 2  4-3X2  +  I2  4-3X54- 
4X  104-4+2  X4— 4+12=  ?    Ans.8. 

S-  13x3+8  +  7-^6x84-3x2-^-3x2  +  7 
+  7  +  64-2  — 10  X3—84-5  +  7X4X 
24-44-15x9x2x3-94-3x44-  II 
x84-3=?     Ans.  32. 

6,  14X  2+4x84-2  X34-7  x8-^3  4- 2  X4 
+  84-2x3-24-3x2x2-1-12x9+8 
+  9—6  +  10  +  3  +  2-1-94-3x2+4=  ? 
Ans.  8. 


l6o  FIRST  STEPS  AMONG  FIGURES. 


7.  i2Xii-^3-f8-j-24-8-r-2X3~3-i-2-^4 

X9-f3-T-3X 4-^2  +  8-^2X3+3  4- 12 
X9-r3  — 2  +  4X9+3X2  +  3X2  +  8 
X4X2=?     Ans.  16. 

8.  i9X2  +  4-r3X2+4Xi2  +  8+2-i-2  +  5 

-4-2X3+2x4+12x8+2X3+4+8 

X 9+ 1 +4X3 -4- 5X2  +  8 -2-4X4 
=  ?     Ans.  60. 

9-  9X3-5^2x3+3^24-9X18+9+8 
+  5-^2X3+9  +  8X3-i-  2x3-2x3 
-7-9X4-^2x5  +  2+5x6  +  3=? 
Ans.  18. 
10.   16x4+2+2x3+12-^4X3-^5X4+2 
X4-r-2  — 6+2X5-r-25X  12  +  8+ 12 
+  2-7-4X10+2+7x15-50x3X2 
-J-3-25+5=?     Ans.  5. 
I.  If  4  lemons  cost  7  cents,  what  cost  20 
lemons  ? 

Explanation :  Teach  the  pupils  that  if  a  cer- 
tain quantity  of  anything  cost  a  certain  amount, 

3  times  that  quantity  will  cost  3  times  as  much; 

4  times  that  quantity  will  cost  4  times  as  much, 
etc. 

Solution :    If   4    lemons    cost   7    cents,   20 
lemons,  which  are  5  times  4  lemons,  will  cost 

5  times  7  cents,  or  35  cents. 

See  P.  Ed.,  p.  119. 


FIRST  STEPS  AMONG  FIGURES.  l6l 

2.  If  3  oranges  cost  lo  centg,  how  many 
oranges  may  be  bought  for  30  cents  ? 

Explanation  :  Teach  the  pupils  that  if  a 
certain  sum  of  money  will  buy  a  certain  quan- 
tity, 3  times  that  sum  will  buy  3  times  that 
quantity,  etc. 

Solution  :  If  10  cents  will  buy  3  oranges,  for 
30  cents,  which  are  3  times  10  cents,  you  can 
buy  3  times  3  oranges,  or  9  oranges. 

3.  What  cost  18  spools  of  thread  at  the 
rale  of  2  spools  for  9  cents  ? 

4.  If  2  knives  may  be  bought  for  5  shil- 
lings, what  will  20  knives  cost  ? 

5.  If  2  men  cut  5  cords  of  wood  in  a  day, 
how  many  cords  will  10  men  cut  in  a  day  ? 

6.  If  3  bushels  of  wheat  cost  $6,  what  will 
8  bushels  cost  ? 

7.  If  2  bOshels  of  wheat  cost  $3,  how  many 
bushels  may  be  bought  for  $18  ? 

8.  36  cents  will  buy  how  many  marbles  at 
3  for  4  cents  } 

9.  If  3  boys  can  do  a  certain  work  in  6 
days,  how  many  days  will  it  take  i  boy  to  do 
the  same  work  ? 

10.  If  2  men  can  hoe  a  field  of  corn  in  4 
days,  how  many  days  will  it  take  i  man  to 
do  it  ?  IX 


1 62  FIRST  STEPS  AMONG  FIGURES. 

11.  If  3  n^^n  can  cradle  6  acres  of  grain  in 
a  day,  how  many  acres  can  i  man  cradle  in  a 
day  ? 

12.  If  2  men  can  build  a  wall  in  6  days, 
how  many  men  can  build  it  in  i  day  ? 

13.  If  4  men  can  dig  a  ditch  in  12  days,  how 
many  men  can  dig  it  in  i  day  ? 

14.  If  2  men  can  dig  8  rods  of  ditch  in  r 
day,   how  many  rods  can  i  man  dig  in  a  day  ? 

15.  If  3  men  can  dig  a  ditch  in  12  days,  how 
many  days  will  it  take  4  men  ? 

Call  attention  of  pupils  to  the  difference 
between  the  15th  example  and  the  i6th.  and 
teach  them  to  find  about  i  of  the  kind  the 
question  asks  about.  For  instance  the  15th 
asks  about  4  men,  hence  find  out  how  many 
days  it  will  take  i  man.  In  the  i6th  it  asks 
about  6  day.s,  hence  find  about  i-day? 

Solution  of  15th  :  If  3  men  can  dig  it  in 
12  days  it  will  take  i  man  3  times  12  days,  or 
36  days  ;  and  four  men  can  dig  it  in  i  of  12 
days,  or  3  days. 

16.  If  3  men  can  dig  a  ditch  in  12  days, 
how  many  men  can  dig  it  in  6  days? 

Solution  :  If  3  men  can  dig  it  in  12  days,  to 
dig  it  in  i  day,  it  will  take  12  times  3  men  or 
See  P.  Ed.,  p.  120. 


FIRST  STEPS  AMONG  FIGURES.  163 

36  men  ;  and  to  dig  it  in  6  days  it  will  take  1-6 
of  36  men,  or  6  men  ? 

17.  A  boy  lost  4  marbles,  then  bought  6,  and 
Josing  10  he  has  35  ;  how  many  had  he  at  first  ? 

18.  What  number  multiplied  by  3  will  give 
12? 

19.  What  number  subtracted  from  7  will 
leave  4  ? 

20.  How  many  days  will  it  take  8  men  to  do 
a  work  that  requires  6  men  12  days  ? 

21.  How  many  men  will  do  a  work  in  25 
days  that  takes  5  men  10  days  ? 

22.  What  cost  9  suits  of  clothes  at  $14  for 
each  coat,  $2  for  each  vest,  and  $4  for  each 
pair  of  pants  ? 

23.  How  many  oranges  at  6  cents  each  can 
be  bought  for  4  cents  and  5  lemons  at  4  cents 
each  ? 

24.  A  boy  gave  10  marbles  worth  7  cents  for 
3  figs  worth  2  cents  each  ;  how  much  did  he 
lose  ? 

25.  What  cost  60  eggs  at  12  cents  a  dozen  ? 

26.  A  boy  has  33  cents,  how  many  marbles 
at  3  cents  each  can  he  buy  and  keep  6  cents  1 

27.  A  boy  has  39  cents,  how  many  must  he 
earn  that  he  may  buy  a  dozen  oranges  at  4 
cents  each  ? 


1 64  FIRST  STEPS  AMONG  FIGURES. 

28.  In  Mary's  garden  are  8  roses,  twice  as 
many  pinks  and  a  dozen  daisies ;  how  many 
flowers  in  her  garden? 

29.  If  3  pounds  of  sugar  cost  24  cents,  what 
will  half  a  pound  cost  ? 

30.  Mary  has  8  cents,  her  sister  has  6  cent.% 
and  their  brother  has  half  as  much  as  both  of 
them  ;  how  many  have  the  three  children  ? 

31.  If  I  buy  60  chickens  at  the  rate  of  5  for 
$2,  and  sell  them  at  the  rate  of  12  for  $5,  how 
much  will  I  gain  ? 

32.  How  far  apart  will  2  men  be  in  7  hours, 
if  they  start  from  the  same  place,  and  travel  in 
opposite  directions,  one  6  miles  an  hour  and 
the  other  4  miles  an  hour  ?  How  far  if  they 
travel  in  the  same  direction  ? 

33.  A  man  who  drives  9  miles  an  hour  is 
trying  to  overtake  a  man  who  is  24  miles  ahead 
of  him  and  who  goes  6  miles  an  hour ;  in 
how  many  hours  will  he  overtake  him  1 

34.  How  many;  ducks  at  the  rate  of  7  for  $6 
can  I  buy  for  $29  and  have  $5  left  ? 

The  pupils  should  mark  each  answer,  and 
also  its  denomination.  They  should  be  required 
to  mark  not  only  the  denomination  of  each  re- 
sult in  the  process  of  solving  problems,  but 


FIRST  STEPS  AMONG  FIGURES.  165 

what  it  represents,  that  is  whether  it  is  cost, 
selling  price,  gain  or  loss,  A's  number,  B's 
number,  &c.  In  this  way  they  will  succeed 
with  many  problems  on  which  they  would 
otherwise  fail. 


EXAMPLES    FOR    THE    SLATE. 

I.  A  had  $8,948  to  which  he  added  $2,284, 
and  then  he  lost  $1,632  when  he  used  all  he 
had  in  buying  38  village  lots  ;  how  much  did 
each  lot  cost  ? 

2.  B  bought  265  acres  (of  land)  for  $22,- 
790  ;  sold  169  acres  of  it  at  $97  an  acre  and 
the  rest  at  cost.     Whole  gain  ? 

3.  A  horse  and  16  oxen  are  worth  $1439* 
and  the  horse  is  worth  $175  ;  what  are  the 
oxen  worth.     What  is  each  oxen  worth  1 

4.  Paid  36  barrels  of  flour  for  60  yards  of 
cloth  at  $6  a  yard  ;  how  much  was  the  flour  a 
barrel  ? 

5.  If  the  front  and  rear  walls  of  a  house 
each  contain  37,390  bricks,  and  the  other  two 
walls  each  49,758;  how  many  bricks  in  the 
four  walls  1 

6.  If  15  boys  walk  900  miles  in  60  days, 
how  far  will  they  walk  in  2  days  ? 

Sec  P.  Ed.,  p.  125. 


1 66  FIRST  STEPS  AMONG  FIGURES. 

7.  Add  forty-five  million  nine  thousand 
ten,  fifty  thousand  eight  hundred,  nine  million 
nine  hundred  thousand  seven  hundred  nine, 
ninety  million  ninety  thousand  seven,  and  six 
hundred  seventy-eight. 

8.  A  sold  one  horse  for  $185,  another  for 
1 1 65,  and  another  for  $187  ;  what  was  the 
average  price  of  a  horse  ? 

9.  Divide  the  product  of  6580  and  7900 
by  their  sum. 

10.  A  bought  300  acres  of  western  land  for 
1,200  ;  B  bought  275  acres  for  $175  less,  and  C 
125  acres  at  $4  an  acre  ;  how  many  acres  did 
the  3  men  buy  ?     How  much  did  they  pay  ? 

11.  A  grocer  bought  279  pounds  of  butter  at 
27  cents  a  pound  and  98  pounds  at  26  cents  a 
pound  ;  he  sold  the  whole  at  ^2  cents  a  pound  ; 
how  much  did  he  gain  ? 

12.  The  weight  of  a  number  of  hogs  was  as 
follows:  250  pounds,  245  pounds,  260  pounds, 
257  pounds,  273  pounds  and  293  pounds  ;  what 
was  their  average  weight  ? 

13.  A  man  wishes  to  buy  a  piano  for  ^375  ; 
he  lays  up  $5  a  week  for  a  year,  or  52  weeks  ; 
how  much  more  must  he  save  to  get  the  piano  ? 

14.  Two  men  start  from  the  same  place  at 

See  P.  Ed.,  p.  128. 


FIRST  STEPS  AMONG  FIGURES.  167 

the  same  time  and  travel  in  the  same  direction, 
one  at  the  rate  of  35  miles  a  day  and  the  other 
44  miles  a  day ;  how  far  apart  are  they  at  the 
end  of  4  days  ?  How  far  apart  if  they  had 
traveled  in  opposite  directions  ? 

15.  Divide  the  product  of  the  sum  and  differ- 
ence of  364  and  93  by  the  difference  between 
their  sum  and  difference. 

16.  A  farmer  bought  one  cow  for  $34,  another 
for  $43  and  another  for  $61;  what  was  the 
average  price  of  the  cows  ? 

17.  The  product  of  two  numbers  is  1,017,702 
and  one  of  them   is  2,758  ;  what  is  the  other? 

18.  What  is  the  sum  of  seventy  thousand 
nine,  nineteen  thousand  six  hundred  forty-nine, 
nine  million  seven  hundred  thousand,  six 
hundred  thousand  nine  hundred  eight,  fifty 
million  sixty,  and  three  hundred  seventy-nine 
thousand  eight  hundred  ninety-eight? 

19.  The  remainder  is  713,  the  quotient  579, 
the  divisor  2758  ;  what  is  the  dividend  ? 

20.  A  man  bought  5  horses  at  $165  each 
and  6  more  for  $902  ;  what  was  the  average 
price  paid  ? 

21.  A  woman  left  her  four  children  $15,000; 
the  eldest  received  one-half  of  it,  and  the  re- 


t 

168  FIRST  STEPS  AMONG  FIGURES. 

mainder  was  divided  equally  among  the  othei 
children  ;  what  was  the  share  of  each  ? 

22.  A,  B  and  C  sold  20  village  lots  for  $14,- 
600 ;  A  received  twice  as  much  as  B,  and  B 
$200  more  than  C,  whose  share  was  $3,500  ; 
what  did  A  receive  ?     B  receive  ? 

23.  A  company  of  14  miners  sell  a  mine  in 
1,245  shares  at  $210  per  share;  what  does 
each  receive  1 

1.  49  +  7»o68  +  9,847+958  4-37-f-489+8,- 

956+9843=? 

2.  946  +  378  +  795+8494-696  +  784  +  359 

+  436  +  775+898=? 
3-  547  +  397+484  +  758  +  969+847+958 
+  497+384  +  947+358  +  596=? 

4.  567  +  498  +  948  +  397+846+372+458 

+  796  +  389  +  486+  958  +347  +  598 

5.  123  +  456  +  789  +  987+456  +  321+743 

+  398^+  476  +395+948  +  767  +  496 
324=? 

6.  578  +  397+956  +  789  +  437+496  +  875 

+  749+  658  +976  +  345+876  +  901 

7.  947  +  643  +  358  +  895  +  769  +  576  +  34-8 

+  954+  847  +659  +  438+987+648 
+  326=? 


FIRST  STEPS  AMONG  FIGURES.  169 

8-  756  +  395+468  +  347  +  579  +  943+658 

+  547+  892  +675+487  +  949+673 
+  246  +  987=? 

9.  391+849  +  327+496  +  327+843+659 
+  742+869+  324  +496+932  +  783 
+  468  +  579  +  453=? 

10.  938  +  493  +  745  +  679  +  548  +  987  +  765 
+  899  +  624+  345  +879  +  354  +  497 
+  384  +  947+486+849+435=? 


TEACHERS'  EDITION, 


fXOM.  NO.              ANS-WKR. 

PAOB.  Na         AKsmn. 

126-  1.  5,245. 

7.  6,475  »«•. 

2.  ,\048. 

8.  3.768  ". 

3.507. 

9.  86.754  «••. 

4.  4,800. 

10.  4.a57  »«. 

5.5.880 

11.  7.684  >». 

6.  614. 

12.  45.876  »•. 

7.  fi62. 

13.4,197. 

128-  8.  215.829,875. 

14.  .5.W.S. 

9.  196,070.418. 

1.5.  5.049. 

10.  114.826,392. 

143-   1.  25.280  rds. 

11.  15.600,168. 

2.  218  cts. 

12.  21.260.981. 

3.  37  cts. 

13.  48,.'i81.904. 

4.873. 

14.  .53.084.772. 

5.  10.208  times. 

15.  193.177.236. 

144-  6.  253  sheep. 
7.  1.884.960  ft. 

US-  1.  465.632 '•». 

2.  3ft4.257  »». 

1.  5.7.56 

3.  536.174  >^. 

2.  .5.478. 

4.  6.527.465  «"•. 

3.  .5.472. 

5.  3,756.486  >«. 

4.  Q.3»Z. 

6.  4,603,758  •-•. 

5.6.160. 

130-  7.  1,537.640  >->. 

6.7.139. 

8.  6.4,53.078  '•\ 

7.  5.792.545. 

^.  4..536.028  *^». 

8.  423,881. 

10.  63.502,487  «. 
— 11.4.a57,068K 

9.  589,188. 

10.  438,524. 

12.  7.684.530  ". 

11.  592.658. 

136-  1.  13,.534  ».' 

12.  4.383.818. 

2.  58.412  M'. 

13.  3.708.979. 

3.  34.«V.'r,  :««. 

14.  31.299.917. 

4.  2.4:i2  ■■*. 

145-15.3,740.188. 

5.  3.2W  "•. 

16.  81.897,816. 

6.  4.587  »'». 

17.  Impossible. 

172 


KEY  TO  BEEBE  S 


1.  8,616. 5T»). 

7. 

2.  129,85:i.i>48. 

153-  8. 

8.  48,417.?J<>. 

9. 

4.48.371.475.     • 

1. 

5.  43.079.076. 

2. 

6.561.747.525. 

3. 

7  490.661.4?2. 

4. 

8.  5. 130.  .532. 

5. 

».  755.a54.848. 

6. 

10.  Impossible. 

7. 

11.  5.7«J.a78. 

154-  8. 

146-  1.  4,750.880 »». 

9. 

8.  6,507,400  >^ 

10. 

8.  6,870,567  **. 

11. 

4.  4,688.679. 

12. 

5.  68.705,870  •■'. 

13. 

6.  6,879,069  *■*. 

14. 

7.  74,868,580  »^. 

15 

8.  69,708.570  *^. 

16. 

9.  4.759,368. 

17. 

10.  3.75a  >••«■, 

18. 

11.  3.5?a  »<«. 

1. 

12.  6,374  «««. 

2. 

18.  4.625  "«. 

8. 

14.  7,364  T«». 

4. 

15.  68,450  >.«»•. 

5. 

147-16.  7,056  »»\ 

6. 

17.  65.704  »««. 

155-  7. 

18.  627,304  «^». 

8. 

19.  3,750  «»•. 

9. 

20.  73.604  ««. 

10. 

148-1.  6,489  •w. 

11. 

2.  867  "•. 

12. 

8.869'. 

13. 

4.  7,096  •«. 

14. 

5.  8,079  •»». 

15. 

6.  6.805  «T». 

16. 

149-  7.  65.704  »<». 

17. 

8.  583,607  '»». 

18. 

152-1.  45  fields;  819  A. 

19. 

2.  5  horses. 

20. 

8.  $19,458. 

21. 

4.  $1,936. 

22. 

5.  40,142,  155. 

23. 

6.  545  ft. 

24. 

$5,430. 

$304  "^ 

18  yds. 

847. 

7.600. 

9,680. 

658. 

5.961. 

7,841. 

803. 

7,568. 

9,066. 

1,005. 

9,881. 

9.687. 

7,416,461. 

679,177,066. 

7.4?2,789. 

622,970.195. 

Impossible. 

6,181.969,778. 

5,770,5&L 

87,687,040. 

676,147,266. 

43,050,000. 

6,  r21. 200,000. 

33.236,122.000. 

583,059,063,000. 

6:37,777,020,000. 

6,978,098  '-•. 

8,907,869  ". 

76,986,970  *^, 

53.795,869  «^ 

37,694,850  »-•. 

46,894,789  ", 

49,579,685  *"•. 

6,745  '.»»i. 

3,547  T.OM, 

49,372  •«. 

476 «.'". 

11,946  «.»w, 

597,860  «. 

1,908,000  «.'*>• 

684  «.«»i, 

789  «•. 


FIRST  STEPS  AMONG  FIGURES. 


73 


25.  768.4S6 '% 

166-  7. 

36.  8,060  »••. 

8. 

27.  7,50S  "». 

9. 

28.  4.598.490,000. 

10. 

29.  42, 00 1.756,  two. 

11. 

80.  75,080  "-. 

12. 

156-81.  870  ««. 

13. 

82.  :^5,387, 180,000. 

14. 

83.  6.890  ". 

167-15. 

*4.  346.857  •  a»«. 

16. 

157-  1.  9,598.752. 

17. 

2.76.160.220. 

18. 

S.  87,653,203. 

19. 

4.  42.5.54.964. 

20. 

5.  11,877,228. 

21. 

6.  0,59'.),496. 

168-22. 

7.  107,8?J,736. 

23. 

158-  8.  71>8,91>0.680  »>». 

1. 

9.  7,968,470  •  '2. 

2. 

10.  73.978.596  o-^K 

3. 

11.  6,970,48»)  '^'a. 

4. 

165-  1.  #2,52  "3-. 

5. 

2.  $1,850. 

6. 

8.  «l,26l;  $79. 

7. 

4.  $10. 

169-8 

5.  174,296  bricks. 

9. 

6.  30  mi. 

10. 

14.5,051,204. 
$179. 

8..589  IS<M0-I4>4M, 

700  A.:  $2,725! 
1.988  cts. 
,  263  lbs. 
«U5. 

86  mi.;  316  ml. 
665  >"•«»«. 
«4*i. 
360. 

60,770.524. 
1,597,595. 

$157    f$2,.500each. 
Eldest  $7,500  ;oth'i 
A,  $7,400;  B,  $3,700. 
$18,675. 
37,247. 
6,916. 
7.742. 
7.660. 
7,079. 
9,033. 
9.395. 
10,102. 


10.  11,854. 


End  of  Key  to  Tsachers'  Edition. 


PUPILS' 

EDITION. 

rAOB.  NO.              ANSWER. 

PAQB.  NO.              ANSWCm. 

88-  1.  251. 

34.  317. 

2.272. 

8T-35.  21.752. 

8.271. 

36.  24,853. 

4.239. 

37.  63,545. 

6.  284. 

38.  22,544. 

6.289. 

39.  98,788. 

7.262. 

40.  62,592. 

8.324. 

41.  66,651. 

9.2152. 

42   77,484. 

10.260. 

43.  28,345. 

34-11.  296. 

44.  72.353. 

12.  295. 

45.  62,156. 

18.308. 

46.  63,234. 

14.335. 

1.  46,206. 

15.  314. 

2.  90,698. 

16.325. 

3.  39,606. 

17.  318. 

4.  42,064. 

18.327. 

38-  5.  93.069. 

.  19.  324. 

6.  24,640. 

85>20.  316. 

.     ri,  60,360. 

21.  317. 

8.69,896. 

22.346. 

9.  62,840. 

23.  319. 

10.  96,908. 

24.302. 

11.  60,846. 

25.  345. 

12.  69.369. 

26.307. 

13.  32,341. 

27.297. 

14.  31,203. 

36-28.  325. 

15.  10,243. 

29.328. 

16.  20,418. 

30.308. 

17.  13,028. 

31.293. 

18.  41,302. 

33.323. 

19.  14,032. 

33.308. 

20.  80,218. 

FIRST  STEPS  AMONG  FIGURES. 


75 


21.  13,024. 

15.  .563,135. 

22.  40,132. 

16.  741,027. 

23.  20,312. 

17.  411,303. 

3»-*24.  2.429. 

18.  623,024. 

25.  2,463. 

19.  71,634. 

26.  2,544. 

20.  44,262. 

27.  2,420. 

21.  443,043. 

28.  2,340. 

22.  476,446. 

29.  2,637. 

23.  5.%,467. 

30.  2,917. 

24.  375,r)67. 

31.  2.996. 

49-25.  577,  .567. 

32.  2,3W. 

26.  760,475. 

40-33.  2,967. 

27.  376,737. 

34.  3,178 

28.  875..%5. 

36.  3,184. 

29.  6<56,.567. 

86.  34,892. 

30.  777,6.57. 

37.  36,775. 

31.267,146. 

38.  35.561. 

32.  365,662. 

41-39.  35,673. 

33.  307,177. 

40.  35,680. 

34.  176,51.5. 

41.  4,655. 

35.  3<32,6()7. 

42.  4.5J4. 

36.  421,2.56. 

43.  4,615. 

37.  277,465. 

43^4.  4,300. 

38.  219,173. 

45.  4,522. 

39.  373,676. 

46.  4.8.37. 

40.  37»,074. 

47.  49.044. 

41.  2.57,1.56. 

48.  48.573. 

42.  471,46.5. 

49.  48.874. 

43.  517.774. 

43-50.  49,083. 

44.  76,516. 

61.  51.204. 

45.  311,8.56. 

4T-  1.  4.258. 

46.  677,172. 

2.  4,011. 

47.  376,046. 

8.  4,741, 

48.  .W2,567. 

4.  4,743. 

50-  1.  40,895. 

6.  4,883. 

2.  42,073. 

6.  4.872. 

3.  41,217. 

48-  7.  l.'3.4.%. 

4.  44,286. 

8.  6r).144. 

6.  42,:i08. 

9.86,425. 

51-  6.  44,.')09. 

10.  663,453. 

7.  41,808. 

11.  762.326. 

8.  46,765. 

12.  42.4.'>3. 

9-  46,.558. 

13.  42.3*t. 

10   4.8,803. 

14.  8,636. 

52-11.  4.628. 

176 


KEY  TO  BEEBE'S 


12.  8,462. 

2.  73,276,956. 

33.  4.(»28. 

3.  79,097,578. 

M.  60,248. 

4.  71.178.967. 

15.  6;i906. 

5.  87,819,971. 

16.  68,402. 

6.  3,765.687. 

17.  46,082. 

7.  68.797,784. 

18.  70.492. 

8.  5.748.957. 

19.  127.068. 

9.  73.287,866. 

20.  92,704. 

10.  86,995.768. 

21.  106.926. 

11.80,773,478. 

22.  109.a'i6. 

12.  2G.579.078. 

23.  79..T59. 

«6-13.  7,170,658. 

24.  92.704. 

14.  94,8iS0,045. 

26.  19:^.578. 

15.  860.097,867. 

•26.  214.496. 

16.  47,159,685. 

27.  193,572. 

17.  32,660,068. 

53-28.  213.704. 

18.  6,697,868. 

29.258,144. 

19.  77.995,682. 

80.  71,284 

•20.  90,a59,567. 

31.  139.066. 

21.  77,706,580. 

82  145,704. 

22.  93,730,146. 

33.  105,738. 

23.  a'>,764.946. 

84.  218,712. 

24  70,299,847. 

86,  219.276. 

25.  77,660,078. 

36.  387,156. 

67-  1.  246,7*2*2. 

37.  231,760. 

2.  290,49*2. 

38.  254,096. 

3.  213,a=i2. 

39.  140.168. 

4.  402,241. 

40.  202.680. 

5.  '249.494. 

41.  182.772. 

6.  281,968. 

42.  -230.175. 

7.  459,704. 

43.  157,824. 

8.  •211.476. 

44.  150,216. 

9.  368,464. 

45.  312,384. 

68-10.  5,480.296. 

60-  1.  52.  ^^79. 

11.  3,595,469. 

2.  45.897. 

12.  5  142.276. 

3.  57.437. 

13.  2,548,23*2. 

4.  50,384. 

14.  2,548,546. 

5.  51,459. 

15.  29.126,064. 

«0-  6.  47,931. 

16.  53,085,480. 

7.  54,123. 

17.  1,483.264. 

8.  54,794. 

18.  1,295.866. 

9.  57,547. 

19.  84.320. 

10.  56,480. 

20.  15,674,618. 

64-  1.  7.867,676. 

21.  1,586,880. 

FIRST  STEPS  AMONG  FIGURES. 


177 


22.  2.i98.402. 

32.  24.ft35  ^^ 

23.  :.'4„V.)8,9i)2. 

m.  3.456.827  »^. 

24.  41.;KW3,168. 

76-1.  63,237. 

25.  4,828,(152. 

2.  61.205. 

28.  313,225. 

3.  63.924. 

27.  4,252.262. 

77-  4.  66,698. 

69-28.  L'2,517.838. 

5.  65,307. 

29.  3,186,721. 

6.  55,344. 

30.  3,430,458. 

7.  52,771. 

31.  11.993,:3ftl. 

8.  57,783. 

82.  27.078..591. 

78-9.  61,074. 

33.  47.923.624. 

10.  60,493. 

34.  2,011,205. 

79-19.  69,274. 

35.  16.799,022. 

20.  69.883. 

70-  1.  3.4;,'8. 

21.  68.111. 

2.  2(»4.:^i7. 

22.  71.276. 

3.  820  300. 

23.  67,740. 

4.  .^,060.905. 

80-  1.  17,438,656. 

6.  520,709. 

2.  23.971,7.52. 

6.  1.0G«».5H)7. 

?,.  •",'  -"'-"i:.'. 

7.  8,O.'>0,907. 

4.       :>. 

8.  64,08;i,207. 

•'». 

9.  -209.071,205. 

0.  ;,'i;.irj.:).5W. 

10.  7.062..3t)4. 

:.  2,59,5,186. 

11.  508,207. 

8.  4.188,375, 

12.  53,208,409. 

9.  5,965.938. 

71-13.  2,4.')7»». 

10  3.235.848. 

14.  3,543  '  *. 

11.  5.5.044,.5.'>5. 

16.  6.475  »». 

12.  17.508.384. 

16.  274,658  >». 

^3.  26,4.35  ^■*. 

17.  545,734  ". 

14.  i].%:2n  a-». 

18.  265,435  »». 

15.  ;{54.(;24  > «. 

19.  238,636  ". 

16.  3.74G.254  »^. 

20.  1,887,677  ''. 

17.  (157.;U2  >*. 

21.  6,543.^46  >"•. 

18.  4.(kJ.-).246  «■». 

22.  64,764  »^». 

81-19.  7.502.4.35**. 

23.  154,264  ♦^. 

'Al  437.520  *•'. 

24.  (M.357  >•*. 

.21.  4..5.37,264»-». 

26.  623.542  «. 

22.  5  743J.'63  «-«. 

26.  75.246  ". 

2;j.  5,762,474  ". 

27.  543.452  *\ 

24.  2,6t»5  »•*. 

as.  974,058. 

^.  6..370  524  «-». 

29.  642.455  ". 

26.  423.177. 

80.  046,819  »•. 

27.  6,258.086. 

81.  786.538  'V 

28.  3,004.756  ^. 

178 


KEY  TO  BEEBE'S 


29.  4.a>3.:02  ^. 

7.  $12,250. 

"^30.  7  0S6.534  «. 

8.  127  marbles. 

31.  -    '-  *  •'♦\ 

9.  105  ct8. 

/3L'                      «. 

10.  1.781  stept. 

,  3;i.              -*. 

11.  $85. 

1.  laiii  '^ 

12.  7.682  ct8. 

2.  3,624  >«. 

lo.  ^m-Jt 

3.  4.235  «»*. 

14.  92  da. 

4.  4.236  «.•«. 

•1-15.  425  A. 

S2-  6.  5..J42  •*». 

16.  $068 

6.  24.353  »•«••. 

17.  91  marbles. 

7.  645  ' «". 

IX.  *  1.920. 

8.  3.624  «>. 

19.  2,5.092  cU. 

9.  3.425  «««. 

20.  59  bu. 

10.  4.r26  "». 

21.  15  cts. 

11.  3,428  « »»•. 

22.  874  bu. 

12.  3,564  «•'". 

23.  152  bu. 

13.  6,457  «>. 

92-24.  1,784,910. 

14.  54.673  '.»". 

1.  64.482. 

15.  4  756  «.«>«. 

2.  68,035. 

16.  57.643  ««. 

3.  69,530. 

17.  3,745  »«. 

4.  66,904. 

18.  4.576.                       ' 

5.  6><.728. 

19.  74,656  »". 

93-  6.  7L-207. 

20.  46,576  »". 

7.  75,781. 

21.  6  534  ^: 

8.  74.612. 

22.  76,487  » »». 

9.  72,156. 

23.  7.458  «. 

10.  69.201. 

24.  64.786  >•'. 

94-11.3.608.157. 

25  5.867  >. 

12.  a58,280. 

26.  4,758  "*. 

13.  016.178. 

27.  6,785  ••». 

14.  266,078. 

28.  65,847  ***. 

15.  6,050  993. 

29.  47.664  >•»«•. 

16.  391  648. 

83-30.  71,414. 

17.  407,708. 

31.  71.083. 

18.  278.079. 

32.  72,225. 

19.  277,164. 

33.  70.304. 

20.  391.667. 

34.  68.796. 

21.  622.678. 

89-  1.  549  bu. 

22.  578.238. 

2.  370  ct8. 

23.  39.213,566b 

»0-  3.  864  bu. 

24.  670.781. 

4.  174  marbles. 

25.  3,764,877. 

6.632  bu. 

26.  79.088. 

6.  189. 

27.  24,147. 

f  IRST  STEPS  AMONG  FIGURES. 


179 


28.  42,869.909. 

-12  5,746,387  ♦-^ 

29.  09.059.978. 

13.  7.468,576  '■•. 

30:  S7J)Zi  970. 

14.  67.580.760  •-». 

81.  3.189.82.5. 

15.  4,0157,586  '-». 

32.  36.970.172. 

16.  65  748.760  »-'. 

1.  70  919.205. 

17.  796.859,809  »-•. 

2.  47.009.1:50. 

18.  097.879,680  »-^ 

9B-  3.  5.449  025. 

19.  04,859  760. 

4.  302.421.261. 

20.  748:695  ♦-^ 

6.  31.269.744 

21.  47i^.508  '-■>. 

6.  392  202.525. 

97-22.  840.-967  »-«. 

7.  3.756,008. 

2:5.  639.-408  T-«. 

8.  726.525. 

24.  4.253  "«. 

9.  187.044. 

2.5.  Omit  this  example. 

10.  7.-221.816. 

26.  35,246  »<". 

11.  5.498.2r>5. 

27.  5,264  3'". 

12.  5.448,946. 

28.  6,423  »•"». 

13.  4,231.810. 

29.  763  '»"^ 

14.  35,677.072. 

30.  4.536  «.o". 

15.  04,:W7.97S. 

31.64,:552"'. 

16.  49.849,081. 

32.  48,372  '". 

17.  610,079,011. 

33.  57,362  s"'. 

18.  308.940.670. 

34.  41,572  so*. 

19.  2H.().K1,4(>4. 

35.  3,754  i-<««'. 

20.  379.514.76.3. 

36.  4.827  "«. 

21.  35  69.5,072. 

37.  36,472  "^ 

22.  68.097.323. 

38.  64  727  *••«. 

23.  68,390,182. 

39.  463.7.52  ♦.»«•. 

24.  379,239,951. 

40.  74,635  "». 

26.  6.257.507,544. 

41.  4  605  "». 

26.  6.328,767. 

42.  70,5:54  »  •«». 

96-27.  2.816.929. 

43.  4,6.53  '.'". 

28.  708.709. 

44.  64,075  "•. 

29.  Impossible. 

45.  736,502  »". 

1.  463.067  »^ 

2.  684,756  ". 

98-46.  47.5,630 '•»♦•. 

47.  4,607  »-^«'. 

8.  475.307  ' «. 

48.  60.835  »«. 

4.  6.463,060 « ». 
6.  6Wite7  '  •. 

49.  72,506  "». 
60.  43.072  •<»•. 

6.  374,675  •  •. 

51.  4,073  »o»<». 

^7.  466,738  »■•. 

62.  47,280  «". 

-  8.  6,537,645  •«. 

63,  58.240  »««. 

9.  6,870,657  •  ^ 

64.  47,060  ". 

10.  4,867,680  » ». 

56.  70,848  «."•. 

11.  6,678  648  »•. 

66.  4,766  »". 

i8o 


KEY  TO  BEEBE'S 


67.  697  »«. 

30.  86  vrs. 

68.  758  **\ 

31.  840.413,978. 

59.  579 ««. 

106-1.  79,984. 

60  796»«. 

2.  76.908. 

61.  975  »«. 

3.  72.1116. 

62.  768  ««. 

4.  87,047. 

63.  687  »*'. 

5.  89,229 

m.  876  »••. 

lot-  6.  74.077. 

65.  8.607  >«. 

7.  81,212. 

91M6.  6.908  '«*. 

8.  76  464. 

e7.  8.007  «'•. 

9.  81.195. 

68.  7.906  «». 

10.  75,826. 

68.  47.660  »•. 

108-11.  79,456. 

70.  76,306  >•«. 

12  82,270. 

71.  67,067  ^»^. 

13.  84,588. 

ri.  67,642  '•». 

14.  78,351. 

102-  1.  ♦1,483. 

15.  78,548. 

2.  728.525. 

100-16.  95,050. 

3.  7,502. 

17.  97,295. 

4.  7  bbl.;  4  gal. 

left. 

18.  97,187. 

5.  684  jral. 

6.  t6,8i5. 

10.  98.616. 

20.  97,489. 

7. 197  mi. 

110-1.  4,246,909. 

8.  $661. 

2.  8,737,469. 

9.  7.126. 

3.  389,385  828. 

103-10.  426. 

4.  73,165,608. 

11.  $1,060. 

5.  29,730,808. 

12.  109  eheep. 

6.  7.080,807. 

18.  Lost  $900. 

7.  291,864.906. 

14. 1.796,266  apples. 

8.  77.927,921. 

15.  $4,543. 

9,  730,868.068. 

16.  59.888.884. 

10.  20,679,828. 

17.  136  tons. 

11.  7.779,807. 

18.  337,022. 

12.  Impossible. 

104-19.  $19. 

13.  39,299.876. 

20.  191,284  yd. 

14.  421.769,927. 

21.  Lost  $50. 

rrcm. 

15.  670,796.919. 

22.  14, horses  and  $50 

16.  6.479.979,692» 

23.  $384. 

17.  4,585,944. 

24.  421  A.;  $37,309. 

18.  5,955,446. 

25.  $6  »« '». 

19.  1,.540,080. 

26.  $4,456. 

20.  772.926.042. 

105-27.  3.743,520  ft. 

^ 

21.  379,239,951. 

28.  as  mi. 

22.  550,842  699. 

29.  NotbiuK. 

23.  6,953,1 40.080i 

FIRST  STEPS  AMONG  FIGURES. 


i8i 


\U.  490,019,928. 

24.  147,283  '■'^. 

25.  4.311.281,464. 

25.  6,485  "0 

1 1 1-2U.  78,(>.-iO,000. 

26.  46.372  •". 

27.  5.730. 

114-27.  14,735  ♦••. 

28.  0.800. 

28.  17.399  "'. 

29.  0,320. OtK). 

29.  68.591  ''*". 

.JO.  87..-)O0.O00. 

30.  147.904  M«. 

112-31.  172..')OO.0O0. 

31.  t».008  *■"*. 

32.  10.205.000. 

32.  13,508,"*. 

33.  444..500.000. 

;«.  49,807  »<». 

M.  18.700,000,000. 

34.  1.38,709  »•»••. 

35.  4,XW,4OO.0O0. 

1.  79  »•«". 

30.  17.983  000. 

2.  are »«. 

37.  51.300,000. 

3.  23  "••00. 

38.  6.887.200. 

4.  702.196  ♦«. 

<».  59,073,000.  : 

5.  37  a»38««. 

40.  40.^43,000. 

116-1   486 '.»•». 

41.  3.407.4;%4.000. 

2.  6.a57  "1". 

42.  452.27(5,040  000. 

3.  790  ".»<><>. 

43.  3,5^9.038,200,000. 

4.  90  «.»", 

44.  4.827.581,000. 

6.  873  «•"•'. 

45.  6,409.800.000. 

6.  98  so.ooo. 

^  1.  4.798.687  *-». 

7.  468  '.'oo. 

2.  87.806,790  '-*. 

8.  2  005,600,000. 

113-3.  709.589,079  »-8. 

9.  7,968  »«». 

4.  9.687.890  a-^ 

10.  68,050  ^M. 

5.  769  809,780  ••. 

11.  7,960  ». 

6.  6,589.679  »^. 

12.  5,920.005.998. 

7.  978,697.089  •». 

13.  48.690  «>■"». 

8.  76.808,697  *-«. 

14.  79.080  TOO. 

9.  7.049,680  » •^ 

llT-15.  1.785  TO"'. 

10.  4.906,704  T-». 

16  60.970". 

11.  79.684,796  «-^ 

17.  9.780  M. 

12.  958.007,980  *-». 

18  Impossible. 

13.  893,798,400 »». 

19.  740  1*7000. 

14.  65.870,486 «-'. 

20.  8.1K59  "«». 

15.  870.956  •-«. 

21.  0,095  «oo. 

16.  75.680,390  •-». 

22.  40;>.026,600. 

17.  903.780  *^. 

Si.  78,096  «»>. 

18.  9.586,090  »-•. 

24.  790.000. 

19.  89.607.980  •-•. 

1.  947,004. 

20.  5.809.759  *-\ 

2.  707.079. 

21.  4,375  « <«'. 

3.  9,058,032. 

22.  6.736  »»'. 

4.  10,149.216. 

23.  365  «'•. 

6.  749.042,756. 

l82 


KEY  TO  BEEBE'S 


a  7,043,688. 

118-7. 

78.JJ56.244. 

8. 

SSIJ.S.'iS.eie. 

9. 

104.154,369. 

10. 

5,7.'i5.896. 

11. 

1  171  (>::,  764. 

IJ 

;. 

IS. 

;o 

u 

.08. 

If. 

IG, 

17 

,^. 

18 

L 

19. 

i,it.t. 1^*^,632. 

20.  95  879.820. 

21. 

98.790.948 »-». 

22.  49.688.182  •-». 

23.  68.874,989  «->«. 

24.  87,076.850  '•". 

26.  786.647.997  ►>«. 

28.  869.897.046  >^*. 

27.  87,998,060  »«^««. 

28.  79,684,968  ••". 

29. 

74.869.740  "«. 

30. 

675,846.090  ^»» 

12&-  1. 

$2,010. 

2. 

$34,995. 

8. 

490,000. 

126-4. 

17vr8. 

6. 

236  A.;  $74  '»•.««•. 

a  15,148. 

7. 

$42. 

8. 123,685. 

9. 

38  mi. 

10. 

60  cU. 

11. 

3,553;  3,066. 

12. 

$760. 

1 27-13.  $14,353.  [A  $27 left 

14. 

174  "-"A.;  or  174  A. 

15. 

614  mi. 

16. 

22,999,800,925. 

17. 

$2,250. 

18. 

22,032  solid  ft. 

19. 

$4,340. 

128-20. 

$8,380. 

21.  $80. 

22.  $80. 

23.  $945. 

24.  192  da. 

25.  $322. 

26.  .584  mi.;  64  mi. 

27.  136  bu. 

28.  $45. 

129-29.  118  »•■»»•.         [rem. 

30.  69  horses,  and  $58 

31.  42  times. 

32.  183  bu. 
ai.  $82. 

34.  ?2  weeks. 

35.  581  sheep. 

36.  429  cts. 

37.  388  lbs. 
13<K38. 15  sheep. 

39.  49  animals. 

40.  6  vrs. 

41.  1,440  lbs. 

42.  3.536. 

48.  $772.  [rem. 

44.  6  horses   and   $40 

45.  1.274  bags. 
131-46.  32  teachers  &  $175 

reiiKiinintr. 

47.  14  cows. 

48.  949  cts. 

49.  Gained  80  cts. 

50.  70  half  dimes. 

51.  37  bags ;  518  cte. 

52.  132  birds. 
132-53. 1. 162,568  "••»'^  sec 

54.  $4,730. 
55.48. 
Omit  "more.** 

56.  Lost  $685. 

57.  $86. 

58. 6,784.  [left. 

59.  76  Go's  and  15  men 

60.  27  yds. 
133-61.  1215rd.;  9,051  rd. 

62.  12  yrs. 
63.158. 


FIRST  STEPS  AMONG  FIGURES. 


'83 


64.  8,589, 7aS.         1 902. 
66.  Quo.  105;  rem.  3,- 
66.  2,392. 
13S-L  70,169. 

2.  67.005. 

3.  58.369. 

6.  74,473. 
130-6.76,967. 

7.  81.025. 

8.  73,892. 
9  72,772. 


10.  74,617. 
ISr-ll.  90,190. 

12.  95.621. 

13.  aj.522. 

14.  89.651. 

15.  93.822. 
138-16.  104,643. 

17.  108.075.  . 

18.  98,269. 
139-19.111,157. 

20.  111,236. 


APPENDIX. 


DETAILED   METHODS   IN   ARITH- 
METIC. 

FROM  THE  COURSE  OF  STUDY  PREPARED  FOR  THE  PUB- 
LIC SCHOOLS  OF  SAN  FRANCISCO. 


I.  Lessons  for  Beginners. 

Grubes  Method  in  Number. 

The  following  are  in  substance  some  of  the 
most  important  principles  given  by  Grube  for 
his  method  in  teaching  beginners  to  compre- 
hend numbers  and  their  relations. 

Prituiples. 

**  I.  Each  lesson  in  Arithmetic  must  also 
be  a  lesson  in  language.  The  teacher  must 
insist  on  readiness  and  correctness  of  expres- 
sion. As  long  as  the  language  for  the  number 
is  imperfect,  the  idea  of  the  number  will  be 
defective. 

"  2.  The  teacher  must  require  the  scholar 
to  speak  as  much  as  possible. 

"  3.  Answers  should  be  given  occasionally 

185 


l86  APPENDIX. 


by  the  class  in  concert,  but  usually  by  the 
scholar,  individually. 

**  4.  Every  process  must  be  illustrated  by 
means  of  objects. 

"5.  Measure  each  new  number  with  the 
preceding  ones. 

**  6.  Teachers  must  insist  on  neatness  in 
making  figures." 

ORDER   OF   STEPS. 

First  Step.  Illustrate  the  required  combi- 
nations by  means  of  counters^  such  as  blocks, 
splints,  or  shells,  in  the  hands  of  the  children 
themselves,  and  by  other  objects  in  the  hands 
of  the  teacher. 

Second  Step.  Express  the  same  combina- 
tions on  the  blackboard  or  on  slates  with 
marks. 

Third  Step.  Take  the  same  combinations 
mentally  with  abstract  numbers. 

Fourth  Step.  Practical  problems  in  applied 
numbers. 

HOW    TO   BEGIN. 

J^*  The  time  required  for  the  work  will 
depend  upon  the  age  of  the  children,  as  also 


APPENDIX.  187 


somewhat  upon  their  natural  ability.  Some 
children  may  require  a  year  tO  complete  the 
work  which  others  may  master  in  a  term. 

I.   THE  NUMBER  ONE.     • 

1.  Hold  up  one  counter,  one  hand,  one 
finger,  one  slate,  etc. 

On  your  slate  make  a  straight  mark,  one 
dot,  one  cross,  etc. 

On  the  blackboards  make  one  mark,  one 
dot,  one  cross,  etc.  • 

2.  Place  one  counter  in  the  middle  of  the 
desk  ;  take  it  away ;  how  many  have  you 
left? 

Make  one  mark  on  your  slate  ;  rub  it  out  ; 
how  many  marks  are  left  ? 

3.  Send  the  class  to  the  blackboards  and 
let  them  make  the  mark  for  one  thus,  |  ;  and 
also  the  figure  thus,  1. 

4.  Proceed  very  slowly.  Much  time  should 
be  given  to  those  who  do  not  learn  easily. 

II.   THE  NUMBER   TWO. 

I.  Each  of  you  take  one  counter  and  place 
it  by  itself  on  your  desk  ;  now  take  another, 


l88  APPENDIX. 


and  place  close  to  it ;  how  many  counters 
have  you  ?  ( Require  the  answer  in  a  full  sen- 
tence.) 

Make  one  straight  mark  on  your  slate ; 
make  another  close  to  it ;  how  many  have 
you  now  ? 

Go  to  the  blackboard  :  make  one  mark  ; 
another,  close  to  it  ;  how  many  now  ? 

Clap  your  hands  once  ;  again ;  how  many 
claps  ? 

Rap  on  your  desk  once  ;  again  ;  how  many 
raps  ? 

2.  Counting. — Place  one  counter  on  your 
desk,  *  ;  a  little  way  off  from  the  first  one, 
place  two  counters  close  together,  thus  *  *. 
Count,  on^  two  ;  tico^  one. 

On  your  slates  make  marks  thus,  |  11 ,  and 
count  fonvards  and  backwards. 

3.  Addition. — I.  Place  one  counter  on  the 
desk  ;  place  another  counter  close  to  it ;  how 
many  have  you  now  ?  Ans.  I  have  tivo  count- 
ers. How  many  counters  are  one  counter  and 
one  counter  ?  Ans.  One  counter  atul  one 
counter  are  tivo  counters.  [The  teacher  will 
further  illustrate  with  books,  pencils,  crayons, 
etc.] 


APPENDIX.  189 


II.  Shte  and  Blackboard. — Make  one  mark  ; 
another  one  near  it.  How  many  marks  have 
you  made  ? 

[Continue  with  rings,  dots,  crosses,  etc.] 

4.  Subtraction. — I,  Place  two  counters  to- 
gether on  your  desk  ;  take  one  away  ;  how 
many  have  you  left  ?  Ans.  I  have  one  left. 
One  counter  from  two  counters  leaves  how 
many  ?  Ans.  .One  counter  from  tuo  counters 
leaves  one  counter. 

[Teachers  will  continue  with  fingers,  hands, 
books,  and  other  objects.] 

II.  Slate  and  Blackboard.  — ^Lakt  two 
marks ;  rub  out  one  ;  how  many  are  left  ? 
Make  two  marks  ;  rub  them  out ;  how  many 
are  left  ?  Ans.  None  are  left.  Two  taken 
away  from  two  leaves  how  many  ? 

5.  Multiplication. — I.  Each  of  you  put  one 
counter  on  the  desk  ;  now  put  another  one 
with  it  ;  how  many  times  have  you  taken  one 
counter  ?  Ans.  I  have  taken  one  counter  twice. 
Two  times  one  counter  are  how  many  count- 
ers ?     Ans.    Tivice  one  counter  are  tico  counters. 

II.  Slate  and  Blackboard. — Make  one  mark ; 
now  another.  How  many  times  have  you 
made  one  mark  ?    Ans.  I  have  made  one  mark 


190  APPENDIX. 


twice.  Then  two  times  one  mark  are  how  many 
marks  ?  Am.  Two  times  one  mark  are  tivo 
marks. 

6.  Division. — See  Teacher's  Edition  of 
First  Steps.  "^—  T4,  paragraph  10. 

7.  Compansofi. — Give  one  counter  to  John 
and  two  to  Frank.  How  many  counters  has 
John  ?  Frank  ?  How  many  has  Frank  more 
than  John  ?    How  many  more  is  two  than  one  ? 

How  many  counters  has  John  less  than 
Frank  ?  Then  one  is  one  less  than  two,  and 
two  less  than  two  is  nothing. 

Blackboards.  —  Illustrate  the  same  with 
marks. 

General  Remarks. 

It  is  a  feature  of  this  method,  that  it  teaches 
by  the  eye  as  well  as  by  the  ear,  while  in 
most  other  methods  arithmetic  is  taught  by 
the  ear  alone.  If  a  child  is  to  measure  7  by 
the  number  3,  the  illustration,  by  comparison 

is  : 

•  *  *  * 

♦  *  * 

* 


APPENDIX.  191 


".If  counters  are  arranged  in  this  way,  and 
impressed  upon  the  child's  memory  as  depict- 
ing the  relation  between  the  number  3  and  7, 
it  is,  in  fact,  all  there  is  to  know  about  it. 
Instead  of  teaching  all  the  variety  of  possible 
combinations  between  3  and  7,  it  is  sufficient 
to  make  the  child  keep  in  min.d  the  above 
picture.  The  first  four  rules,  as  far  as  3  and 
7  are  concerned,  are  contained  in  it,  and  will 
result  from  expressing  the  same  thing  in  dif- 
ferent words,  or  describing  the  picture  in  dif- 
ferent ways.  Looking  at  the  picture,  the 
child  can  describe  it  as  : 

3 -f  3  +  1  =  7,  or3X  2  4- 1  =  7,  or  7 -3-3=  I, 
7-7-3  =  2  (i).  The  latter  process  is  to  be 
read  :  3  in  7  twice,  and  i  remaining. 

"  Let  the  number  to  be  measured  be  10, 
and  the  number  by  which  it  is  to  be  measured 
be  4  ;  then  the  way  to  arrange  the  dots  is  : 

♦  ♦  ♦  * 

*  *  *  * 

"  The  child  will  be  able  to  see  at  once,  by 
reading,  as  it  were,  that  4-1-44-2  =  10,  4x2 
-h2  =  io,  10  —  4—4=2,  io-f-4  =  2  (2),  and  to 


192  APPENDIX. 


perceive  at  a  glance  a  variety  of  other  com- 
binations. The  children  will,  in  the  course 
of  time,  learn  how  to  draw  these  pictures  on 
their  slates  in  the  proper  way.  Nor  will  it 
take  long  to  make  them  understand  that  every 
picture  of  this  kind  is  to  be  *  read  *  in  four 
ways,  first  using  the  word  and^  then  times^  then 
/esSy  then  in.  As  soon  as  the  pupils  do  this, 
they  have  mastered  the  method,  and  can 
work  independently  all  the  problems,  within 
the  given  number,  which  are  required  in  meas- 
uring." 

Ord^r  of  Steps. 

I.  Counters. 

II.  Figures. 

III.  Abstract  Numbers. 

IV.  Practical  Problems. 


FIRST    STEPS 

Among    Figures. 

A  Drill  Book  in  the  Fundajnental  Rules 
of  Arithmetic. 


Pupils'  Edition. 


BY 

LEVI    N.    BEEBE 

CANANDAIGUA,    N.    Y. 


SYRACUSE.    N.   Y  : 

C.  W.   Bardebn,   Publisher. 
1881. 


CormoHT.  1877,  L»n  N.  B»«Ba. 


PREFACE  TO  REVISED  EDITION. 

Great  care  has  been  taken  to  correct  the 
errors  of  the  first  edition  and  it  is  hoped  that 
few  remain. 

At  the  request  of  several  teachers  in  un- 
graded schools,  the  addition,  subtraction,  mul- 
tiplication, and  division  tables  have  been  in- 
serted in  the  Pupils'  Edition,  and  a  number  of 
pages  have  been  prefixed  to  relieve  teachers  in 
such  schools  of  much  of  the  labor  of  oral  in- 
struction. In  graded  schools  the  pupils  should 
not  have  a  book  until  they  liave  been  taught 
orally  as  far  as  to  the  tables  of  y's  and  review, 
p.  29. 

'I'lie  Teachers'  Edition  contains  the  answers 
to  the  examples  in  this  book,  and  instruction 
for  oral  work  with  the  youngest  pupils,  together 
M  ith  methods  and  additional  examples.  It  is, 
therefore,  necessary  that  the  teacher  should 
have  it  and  carry  along  the  work  of  the  two 
editions  together.  References  at  the  bottom 
of  the  pages  in  each  edition  call  attention  to 
the  pages  of  the  other  edition  which  contain 
work  of  the  same  kind. 

The  Pupils'  Edition  is  bound  separately  for 


!v  PREFACE  TO  REVISED  EDITION. 

pupils'  use,  as  well  as  with  the  Jl'eachers*  Edi- 
tion for  teachers'  use. 

The  object  of  having  the  double  book,  the 
Teachers'  Edition  and  the  Pupils  Edition,  is 
that  while  the  pupil  is  to  ])rej)are  his  lesson 
from  the  Pupils'  Edition,  the  teacher  has,  in  the 
Teachers*  Edition,  additional  examples  which 
the  pupil  has  not  seen,  which  are  intended  to 
be  assigned  for  solution  during  recitation  as  a 
test  of  his  knowledge  of  the  subject  in  hand. 

The  first  96  pages  of  the  Teachers'  Edition, 
together  with  the  parallel  work  of  the  Pujjils' 
Edition  to  p.  44,  are  bound  separately  for  the 
use  of  teachers  in  the  oral  instruction  of  pupils 
during  their  first  two  or  three  years  in  graded 
schools. '  It  is  intended  wholly  for  oral  work 
and  is  called  "  First  Steps  Among  Figures,  Oral 
Edition." 

Much  care  has  been  taken  in  each  edition  to 
proceed  from  the  easiest  examples  to  those  that 
are  more  difficult,  in  order  to  avoid  discourag- 
ing the  pupil. 

The  author  is  indebted  to  his  assistant  teacii- 
ers  for  aid  in  the  preparation  of  the  very  large 
number  of  examples  in  the  book. 

For  a  more  extended  notice  of  the  scope  and 
plan  of  this  work  see  the  preface  to  the  Teach- 
ers' Edition,  and  also  the  Special  Notice  which 
precedes  it. 

LEVI  N.  BEEBE. 

Canandaigua,  N.  Y.,  April,  1878. 


SPECIAL  NOTICE. 

Persons  wlio  may  examine  this  book  are 
asked  to  notice  especially  the  following:  Ex- 
amples for  rapid  solving  on  pp.  17,  20,  25,  29, 
38,  41,  45,  51,  etc.,  as  well  as  the  foot  notes, 
pp.  16  and  17;  the  drill  in  reading  and  writing 
numbers  on  pp.  39,  47,  53,  60,  69,  83,  in  to 
116,  etc.;  the  series  of  division  with  remain- 
ders, which  is  to  prepare  the  pupil  for  short 
division,  pp.  107,  128,  129,  145,  and  157.  In 
the  examples  in  long  division,  on  p.  136,  Teach- 
ers' Edition,  and  on  p.  82  of  the  Pupils'  Edi- 
tion, since  the  second  figure  from  the  left  in  the 
divisor  is  a  cipher,  while  the  figures  of  the 
quotient  are  small,  the  divisor  is  contained  in 
each  partial  dividend  just  as  many  times  as  it 
appears  to  be.  For  instance,  the  first  divisor 
in  the  Pupils'  Edition,  p.  81,  is  201  ;  and  if  the 
reader  doubts  that  the  path  is  thus  made  easy 
for  the  beginner,  let  him  give  his  pupils  an  ex- 
ample with  19,  29,  or  291  for  a  divisor,  and 
then  one  with  201  as  a  divisor.  There  are  no 
ciphers  in  the  quotients  on  those  pages,  so  that 
every  difficulty  is  postponed  to  a  later  time  that 
can  be  so  put  off.     The  examples  were  made 


VI  SPECIAL  NOTICE. 

by  assuming  such  a  divisor  and  quotient  as 
were  desirable,  multiplying  them,  and  adding 
an  assumed  remainder  to  the  product,  which 
gave  the  dividend  found  in  the  book. 

After  tlie  practice  on  the  twelve  examples  on 
p.  136  in  the  Teachers'  Edition,  and  on  the 
twenty-nine  examples  on  p.  82  of  the  Pupils* 
Edition,  there  are  a  number  of  pages  of  other 
work,  after  which  long  division,  with  the  same 
sort  of  easy  examples,  recurs  on  p.  146,  Teach- 
ers* Edition,  and  p.  97,  Pupils'  Edition.  After 
a  few  examples  the  cipher  occurs  in  the  quo- 
tient, of  which  the  teacher  is  warned  at  the 
bottom  of  p.  146,  Teachers'  Edition.  Addi- 
tional difficulties  are  treated  on  pp.  147  and 
148,  followed  by  examples  in  illustration,  and 
still  others  on  pp.  114,  115,  and  116,  Pupils* 
Edition. 

The  continuous  form  commonly  used  for 
tables  of  addition,  subtraction,  multiplication^ 
and  division,  (as  2  and  2  are  4,  3  and  2  are  5^ 
4  and  2  are  6,  etc.),  has  been  forsaken  for  the 
form  found  on  pp.  ^^,  37,  43,  48,  56,  63,  etc.. 
Teachers'  Edition,  and  on  pp.  6,  9,  18,  24,  29^ 
etc..  Pupils'  Edition. 

If  the  teacher  copies  the  series  on  the  black- 
board, he  may  write  the  answers  underneath, 
or  require  the  pupils  to  find  the  answers,  as  he 
prefers. 

For  instructions  see  pp.  42  and  43,  Teachers^ 
Edition,  and  pp.  5,  6,  and  19,  Pupils'  Edition. 

The  teacher  will  see  that  at  p.  63  he  should 
begin   to   use   also  the  Pupils'  Edition,  even 


SPECIAL  NOTICE.  VU 


though  the  pupil  has  not  yet  obtained  his  book* 
The  number  at  the  bottom  of  that  page  refers 
to  the  page  of  the  Pupils'  Edition  that  has  the 
same  kind  of  work.  The  numbers  at  the  bot- 
tom of  the  following  pages  in  the  Teachers' 
Edition  refer  in  the  same  way,  while  those  in 
the  Pupils'  Edition  refer  back  to  the  Teachers' 
Edition.  The  work  of  the  two  editions  after 
reaching  these  parallel  pages  should  be  carried 
along  together  carefully. 

The  operations  of  addition,  subtraction,  mul- 
tiplication, and  division  being  taught  together 
throughout  the  whole  book,  beginning  with  the 
easiest  examples  and  progressing  gradually  to 
more  difficult  ones,  work  in  each  rule  is  con- 
stantly recurring.  This  necessitates  a  peculiar 
arrangement  of  the  work,  but  it  constitutes  one 
of  its  chief  excellencies. 

The  Pupils'  Edition  is  bound  separately  for 
pupils'  use. 

The  key  containing  answers  will  be  found  on 
pp.  1 71-183,  Teachers'  Edition. 

For  a  still  further  description  of  the  plan  of 
the  book  please  read  the  preface  of  each  edition 


FIRST  STEPS  AMONG  FIGURES. 


Addition  Table,      fs  and  review. 

The  following  table  is  best  learned  by 
repeating  each  set  of  numbers  many  times, 
as:  "2  and  4  are  6,"  "2  and  4  are  6,"  etc., 
until  it  is  well  learned.  ''Oft  repeated, 
long  remembered''  In  reciting,  it  is  best 
not  only  to  say  2  and  4  are  6,  but  also  2 
from  6  leaves  4,  and  so  on  through  the 
table,  or  at  least  until  the  pupil  sees  clearly 
that  subtraction  is  the  opposite  of  addition. 
If  the  table  is  copied  on  the  blackboard 
without  the  answers,  the  recitation  may  be 
conducted  as  in  oral  spelling,  each  pupil 
reciting  a  section,  and  any  error  being  cor- 
rected in  his  turn  by  the  first  pupil  who 
has  noticed  it.  The  teacher  may  ask  the 
questions  directly  from  the  table  in  the 
book,  if  he  prefers  it. 

The  letters  are  names  for  the  sections, 
and  are  a  convenience  in  assigning  the 
lesson. 

When  the  pupils  have  learned  the  fol- 
lowing table,  test  their  knowledge  of  it  by 
the  table  in  Teachers'  Edition,  p.  63,  which 
is  differently  arranged. 


FIRST  STEPS  AMONG  FIGURES. 


A 

^ 

^ 

4 

2 

5 

3     I 

4 

2     5     3 

2 

3 

4 

5       2 

8     3 

3 
7 

4     5     2 

6 

5 

9 

6  10     5 

£/ 

t 

/ 

g 

2     4 

I 

5 

3     I 

4 

2 

5         3     I 

5    4 

3 

2 

3    4 

5 

2 

3         4     S 

784      765      948      76 

"2  and  4  are  6"  may  be  written  2-1-4= 
6,  and  is  read  **2  plus  4  equals  6."  (Al- 
ways recite  from  below  upward,  that  is  2 
and  4,  not  4  and  2). 

1.  George  has  3  cents  and  Mary  has  5 
cents ;  how  many  hav^e  both  ? 

Solution :  They  have  the  sum  of  3  cents 
and  5  cents,  or  8  cents. 

2.  John  has  4  marbles  and  Henry  has 
2  ;  how  many  have  both  ? 

3.  Susan  had  5  pins  and  afterward 
found  4 ;  how  many  had  she  then  ? 

4.  A  good  boy  brought  in  4  armfuls  of 
wood  for  his  mother  in  the  morning,  and  3 
armfuls  in  the  afternoon ;  how  many  arm- 
fuls did  he  bring  in  that  day  ? 


FIRST  STEPS  AMONG  FIGURES. 


5.  Nettie  found  2  eggs  in  one  nest  and 
5  in  another;  how  many  did  she  find  in 
both? 

6.  Helen  had  3  sleigh-rides  on  Monday 
and  3  on  Tuesday ;  how  many  did  she 
have  in  the  two  days  ? 

7.  Carrie  had  I  needle  and  her  mother 
gave  her  5  ;  how  many  had  she  then  ? 

8.  Charles  rode  down  hill  on  his  own 
sled  4  times  and  on  his  brother's  sled  4 
times ;  how  many  times  did  he  ride  down 
hill? 

9.  Count  by  2's  from  2  to  10,  thus:  2, 
4,  6,  8,  10. 

10.  Read  the  following  numbers,  or 
write  them  in  words  and  bring  them  to 
the  recitation :  y^',  94;  58;  85;  6^  \  93; 
72;  25;  61  ;  38;  15;  88;  29;  13;  36; 
11;  21  ;   12;  34;  98;   14. 

1 1.  Count  by  2's  from  2  to  20. 

Write  in  figures  (Arabic):  (12.)  forty- 
six;  (13.)  fifty- three  ;  (14.)  eighty-one; 
(15.)  sixty-nine;  (16.)  thirty-five;  (17.) 
twenty- seven  ;  (18.)  fifty  ;  (19.)  seventeen  ; 
(20.)  twelve.  The  teacher  should  give 
more  exercise  of  this  kind. 

21.  What  cost  an  orange  at  5  cents  and 
a  lemon  at  3  cents  ? 

22.  Walter  bought  one  book  for  2  shil- 

Sce  Teachers'  Edition,  p.  64. 


8  FIRST  STEPS  AMONG  FIGURES. 

linens  and    another   for   4  shillings ;    how 
much  did  he  pay  for  both  ? 

23.  An  inattentive  pupil  whispered  3 
times  in  the  forenoon  and  4  times  in  the 
afternoon ;  how  many  times  did  he  whis- 
per during  the  day  ? 

24.  On  Nellie's  birthday  her  father  and 
mother  each  gave  her  five  shillings ;  how 
much  money  did  both  give  her  ? 

25.  Count  by  2's  from  2  to  60. 

26.  Write  in  figures  (Arabic) :  VII ;  II ; 
IV;   I;  VIII:   VI;   IX. 

27.  Write  in  letters  (Roman):  5;  9; 
2;  4;  7;   10;  3;  6;    I ;  8. 

28.  Count  by  2's  from  i  to  9. 

29.  Count  by  2*s  from  i  to  13. 

30.  Add  2,  2,  I,  2,  2,  I,  2,  2.  The 
teacher  will  teach  the  pupil  to  write  the 
numbers  on  his  slate  in  a  column,  with  a 
line  underneath,  and  to  write  the  answer 
beneath  the  line. 

31.  Add  I,  2,  2,  2,  I,  2,  I,  I,  2,  2. 

32.  Add  2,  2,  2,  I,  2,  2,  I,  2. 

33.  Add  I,  I,  I,  2,  2,  2,  2,  2. 


FIRST  STEPS  AMONG  FIGURES. 


Subtraction  Table.*     fs  and  review. 
"  Oft  repeated,  long  remembered^ 


a 
6     7 
3     2 

4 
3 

b 
8     7 

4     5 

5 

2 

c 
10     5     9 

5     4    5 

3     5 

d 
4    8    7 
2    3    4 

I 

6 

5 

4     2 

e 
5    6 

3    2 

3 

9 
4 

5     I     4 

/                g 
8    3         7     6 
5    2         3     4 

2    5    3         12    4         5    3     1         4     2 

"3  from  6  leaves  3"  may  be  written 
6— -3  =  3,  and  is  read  "6  minus  3  equals  3." 

After  this  table  is  learned,  test  the  class 
by  using  the  table  in  Teachers'  Edition,  p. 
6^,  which  is  difierently  arranged. 

1.  Count  by  2's  from  i  to  19. 

2.  Jesse  having  7  cents  spent  3  of  them  ; 
how  many  had  he  left  ? 

Solution  :  He  had  left  the  difference  be- 
tween 7  cents  and  3  cents,  or  4  cents. 


•  The  teacher  will  treat  this  table  much  as  he  did  the 
previous  addition  table.  See  the  instructions  with  that 
lable. 


to  FIRST  STEPS  AMONG  FIGURES. 

3.  Edward  had  9  cents;  he  lost  four  of 
them  ;   how  many  liad  he  left  ? 

4.  Albert  had  5  marbles,  and  gave  John 
2  of  them  ;  how  many  had  he  left  ? 

5.  William  has  10  cents  and  Lewis  has 
5  cents ;  how  many  more  has  William  than 
Lewis  ? 

6.  Eight  boys  were  skating  on  the  ice, 
when  three  of  them  fell ;  how  many  re- 
mained standing  ? 

7.  Count  by  2's  from  I  to  29. 

8.  Read  the  following  numbers:  100; 
163;  175;  158;  189;  107;  309;  246; 
350;  416;  112;  761;  468;  9C6;  413; 
970;  320;   518;  971;  800. 

Write  the  following  numbers  in  figures 
{Arabic):  (9.)  four  hundred  sixty-three; 
{10.)  two  hundred  thirty- nine ;  (i  i.)  seven 
hundred  eighty-one;  (12.)  one  hundred 
sixteen;  (13.)  five  hundred  eight;  (14.) 
six  hundred  ninety;  (15.)  seven  hundred 
twelve;  (16.)  three  hundred  twenty-one; 
{17.)  five  hundred;  (18.)  nine  hundred 
two;  (19.)  five  hundred  sixty-one;  (20.) 
one  hundred  seven;  (21.)  the  number  of 
days  in  a  year ;  (22.)  eight  hundred  ;  (23.) 
one  hundred  nineteen  ;  (24.)  nine  hundred 
forty. 

25.   Count  by  2's  from  i  to  69. 

See  Teachers'  Edition,  p.  40. 


KiK.^1   STEPS  AMONG  FIGURES.  II 

Write  in  figures  (Arabic):  (26.)  XV; 
427.)  XXII;  (28.)  XIX;  (29.)  XXIX; 
<30.)  XXIII;  (31.)  XVI;  (32.)  IX;  (33.) 
XXIV;  (34.)  XIV. 

Write  in  letters  (Roman) :  (35.)  twelve  ; 
(36.)  nineteen;  (^y.)  fifteen;  (38.)  twen- 
ty-four; (39.)  twenty-two;  (40.)  twenty- 
five;   (41.)  twenty-three. 

42.  Count  by  3's  from  3  to  15. 

43.  Add  I,  2,  2,  2,  2,  2,  I,  2,  2,  2. 

44.  Add  2,  I,  2,  I,  2,  2,  I,  2,  2,  2. 

45.  Add  2,  2,  2,  I,  2,  2,  2,  2,  2,  2. 

46.  Add  2.  2,  I,  2,  2,  I,  I,  2,  2,  2. 
Make  and  solve  several  examples  like 

the  preceding  ones. 

47.  A  dwarf  tree  had  7  pears  on  it,  but 
4  of  them  fell  oflf;  how  many  remained  on 
the  tree  ? 

48.  Samuel's  mother  gave  him  8  cents 
to  spend ;  he  paid  4  cents  for  candy,  and 
bought  marbles  with  the  remainder ;  how 
much  did  he  spend  for  marbles  ? 

49.  Carlos  started  for  school  with  9 
marbles  in  his  pocket,  but  when  he  got 
there  he  found  he  had  lost  all  but  4  of 
them  ;  how  many  had  he  lost? 

50.  Maggie  had  6  needles;  she  broke 
3  of  them  ;  how  many  were  unbroken  ? 

51.  How  many  more  wheels  has  a  car<. 
riagc  than  a  sulky  ? 


12  FIRST  STEPS  AMONG  FIGURES. 

52.  How  many  less  fingers  on  one  hand 
than  on  both  ? 

53.  8  is  how  many  more  than  3  ? 

54.  Mr.  Jones  was  idle  3  days  of  a 
week ;  how  many  days  of  the  week  did  he 
work  ? 

55.-  Mr.  Rawson  worked  5  days  one 
week  and  4  the  next ;  how  many  days  did 
he  work  in  the  two  weeks  ? 

$6.  Julia  misspelled  2  words  in  the  fore- 
noon and  4  in  the  afternoon ;  how  many 
did  she  misspell  that  day? 

57.  Joseph's  mother  told  him  he  mighi 
eat  2  apples ;  he  ate  5  apples ;  how  many 
more  did  he  eat  than  he  ought  ? 

58.  From  Mr.  Collins's  house  to  the 
post-office  it  is  4  miles,  and  from  the  post- 
office  to  the  school-house  it  is  3  miles  far- 
ther; how  far  is  it  from  Mr.  Collins's 
house  to  the  school-house  ? 

59.  Count  by  3's  from  3  to  24. 

60.  Add  I,  2,  2,  2,  2,  3,  3,  2, 

61.  Add  2,  2,  2,  I,  2,  2,  2,  2, 
2,  2. 

62.  Add  I,  2,  2,  I,  I,  2,  2,  I, 

2,  2. 

63.  Add  I,  2,  2,  2,  I,  2,  2,  2,  I,  I,  1,2, 
2,  2. 

64.  Add  2,  2,  2,  2,  I,  2,  2,  2,  2,  2,  I,  I, 
I,  2,  2. 

Sec  Teachers'  Edition,  p.  66. 


2, 

2. 

2, 

2, 

2, 

2, 

2, 

2, 

2, 

I, 

FIRST  STEPS  AMONG  FIGURES.  1 3 

65.  Add  I,  2,  I,  2,  3,  3,  3,  I,  2,  3,  3,  2, 
I,  2,  2. 

66.  Add  2,  2,  I,  2,  2,  2,  2,  2,  3,  3,  2, 
2,2. 

67.  Susan  took  3  peaches  from  a  pile 
of  6  peaches ;  how  many  were  left  in  the 
pile  ? 

68.  A  boy  was  sent  to  the  store  with 
8  cents ;  he  spent  all  of  them  but  3  ;  how 
many  did  he  spend  ? 

69.  William  spent  2  cents  for  nuts  and 
5  cents  for  licorice;  how  much  did  he 
spend  altogether? 

70.  John  hit  James  4  times  with  snow- 
balls and  James  hit  John  twice ;  how 
many  more  times  was  James  hit  than 
John  ? 

71.  In  a  street  car  are  4  women  and  5 
men  ;  how  many  persons  in  the  car? 

72.  There  were  2  pictures  on  one  side 
of  a  wall  and  3  on  the  other ;  how  many 
on  both  ? 

73.  James  had  8  doves ;  2  pairs  of  them 
were  killed  ;  how  many  lived  ? 

74.  A  little  girl  has  a  cushion  with  6 
pins  on  it ;  she  took  off  3  of  them ;  how 
many  were  left  ? 

75.  A  little  boy  earned  2  cents;  he 
then  had  7  cents;  how  many  had  he  at 
first? 


14  FIRST  STEPS  AMONo  ti»..UKES. 

76.  In  a  geography  class,  4  pupils  were 
in  order  and  3  were  not  in  order;  how 
many  pupils  in  the  class  ? 

When  a  number  to  be  read  contains 
more  than  three  figures,  place  a  comma 
before  the  third  figure  from  the  last ;  thus, 
3,216.  It  is  read,  three  thousand  two 
hundred  sixteen. 

Read  the  following  numbers:  (i.)  5281  ; 
(2.)  6157;  (3.)  9640;  (4.)  7500;  (5.) 
8609;  (6.)  2050;  17.)  7008;  (8.)  1000; 
(9.)  7318;  (10.)  5071;  (II.)  4019;  (12.) 
2800;  (13)  5000;  (14.)  .8060;  (15.) 
7801  ;  (16.)   1007. 

The  teacher  should  show  the  pupil  that 
in  writing  a  number  that  contains  thou- 
sands a  comma  should  be  placed  after  the 
number  of  thousands,  and  then  the  remain- 
der of  the  number  should  be  written  at 
the  right,  any  of  the.  three  places  which 
are  not  occupied  being  filled  with  ciphers. 

Write  in  Arabic:  (17.)  three  thousand 
six  hundred  forty-five;  (18.)  seven  thou- 
sand nine  hundred  fifty-one;  (19.)  one 
thousand  two  hundred  seventy;  (20.) 
eight  thousand  three  hundred;  (21.)  two 
thousand  twenty;  (22.)  six  thousand  one 
hundred  five ;  (23.)  nine  thousand ;  (24.) 
five  thousand   seven;  (25.)   one  thousand 

See  Teachers'  Edition,  p.  53. 


FIRST  STEPS  AMONG  FIGURES.  1 5 

four  hundred;  (26.)  seven  thousand  nine 
hundred  four  ;  (27.)  five  thousand  forty ; 
^28.)  eight  thousand  six ;  (29.)  four  thou- 
sand. 

30.   Count  by  3's  from  3  to  60. 

Write  in  Arabic :  (31.)  XXXVI;  (32.) 
XXIV;  -33  )  LV;  (34.)  XL;  (35)  LXII; 
{36.)  LXXXVII;  (37.)  LXIX;  (3S.) 
LXXVI ;  (39.)  XIX. 

Write  in  Roman:  (40.)  twenty-eight; 
(41.)  fifty-four;  (42.)  forty-eight;  (43.) 
seventy-seven;  (44.)  eighteen;  (45.)  four- 
teen;  146.)  eighty-six;  (47.)  thirty-two; 
(48.)  sixty-nine;  (49.)  fi fty- seven ;  (50.) 
sixteen;  (51.)  thirty-four;  (52.)  seventy- 
five;  (53.)  forty-seven;  (54.)  eighty- 
eight;  (55.)  sixty-nine. 

56.  Count  by  3's  from  I  to  61. 

57.  Count  by  2's  from  2  to  60. 

58.  Count  by  2's  from  I  to  61. 

59.  Count  by  3's  from  3  to  60. 

The  teacher  should  give  a  great  deal  of 
practice  in  counting  as  in  the  four  exam- 
ples above. 

The  teacher  will  show  the  pupil  that  the 
numbers  in  the  following  examples  are  to 
be  so  written  in  columns  that  the  right 
hand  figures  shall  be  in  one  column.  Add 
the  right  hand  column  first.  Write  below 
it  the  right  hand  figure  of  the  result  and 


l6  FIRST  STEPS  AMONG  FIGURES. 

add  the  left  hand  figure  to  the  next  col- 
umn. 

60.  Add  31,  23,  30,  2,  33,  13,  22,  and  2. 

61.  Add  22,  31,  T2,  23,  3,  20,  32,  3, 
and  23. 

62/ Add  33,  20,  2,  13,  33,  30,  22,  23, 
and  30. 

63.  Add  23,  I,  33,  20,  21,  2,  10.  23.  12, 
and  33. 

64.  Add  13,  20,  23,  2,  20,  12,  3.  30,  23, 
and  I. 

65.  Add  33,  30,  2.  23.  20,  33,  3,  23,  10, 
3,  32,  and  12. 

66.  Add  32:  13,  33,  20,  12,  32,  22,  33, 
21,  13,32,  and  33. 

6t.  Add  30,  2,  33,  21,  32,  23,  3,  30,  33, 
21,  3,  and  2. 

68.  Add  23,  33,  20,  12,  3,  32,  30,  I,  13, 
32,  33,  and  23. 

1.  In  a  certain  street  there  were  I  a 
houses,  but  last  night  5  of  thern  were 
burned  ;  how  many  remain  ? 

2.  Fannie  had  5  pinks ;  she  gave  three 
of  them  to  Jane  ;  how  many  did  she  keep  ? 

3.  A  boy  could  not  tell  how  many  5 
and  3  and  2  are.  Can  you  ?  Show  that 
you  can. 

4.  Kate  had  a  dime  ;  she  spent  10  cents 
for  a  doll ;  how  many  cents  had  she  left  ? 

5.  There  were  7  boys  on  a  bench,  5  of 


FIRST  STEPS  AMONG  FIGURES.  1 7 

.whom   were  studying;    how   many  were 
idle  ? 

6.  Fred  had  5  cents  left  after  spending 
4  cents ;  how  many  had  he  at  first  ? 

7.  Four  boys  and  three  girls  brought 
their  lunch  to  school ;  how  many  do  you 
think  stayed  at  noon  ? 

8.  Sherman  had  3  books  given  him  on 
Christmas;  he  had  3  before;  how  many 
has  he  now  ? 

9.  Nine  boys  were  playing  soldier ;  5 
of  them  were  called  home ;  how  many 
were  left  to  play  ? 

10.  8  apples  are  4  more  than  James  has ; 
how  many  has  he  ? 

11.  Harry  jumped  3  feet  and  Jimmy 
jumped  2  feet  farther  ;  how  far  did  Jimmy 
jump? 

12.  A  bad  boy  threw  5  kittens  into  the 
pond  ;  3  of  them  swam  to  the  shore ;  how 
many  were  drowned  ? 

13.  Six  ^boys  were  snow-balling;  two 
of  them  were  hurt  so  they  would  not  play 
any  more  ;  how  many  finished  the  game  ? 

14  Bennie  is  5  years  old  and  Carrie  is 
4  years  older ;   how  old  is- she  ? 

15.  Add  31,  23,  21.  3,  33,  20,  12,  32, 
23,  3,  and  33. 

16.  Add  12,  3,  21,  33,  30,  23,  12,  22, 
20,  32,  and  3. 

S«f  Teacher*'  Eailion.  p.  67. 


l8  FIRST  STEPS  AMONG  FIGURES. 

17.  Add  3,  32,   13,  33,  20,   12,  2,  13, 

22,  31,  23,  and  13. 

18.  Add  31,  23,  33,  2,  13,  22,  30,  33, 

23,  31,  22,  and  33. 

19.  Add  2.  23,  33,  20,   13.  21,  3,  33, 
20,  31,  13.  23,  and  21. 

20.  Add  32,   13,  22,  33.  3,  21,  30,  33, 
13.  22,  33,  and  32. 

21.  Count  by  3*s  from  2  to  62. 

22.  Count  by  4's  from  4  to  20. 

Multiplication  T.able.  fs  and  review, 
"  Oft  repeated,  long  remembered^ 


a 

b 

c 

4 
2 

2 

3 

5 
4 

3 

5 

15 

I 
2 

2 

4 
3 

12 

2     5 
4     5 

3 

2 

8 

6  20 

8  25 

6 

^ 

^ 

2 

5 

10 

4 
4 

16 

I 
3 

3 

5 
2 

3     I 
3     4 

10 

9    4 

4 

5 

/ 

2 

2 

5 
3 

3  I 

4  S 

20   4    15 


FIRST  STEPS  AMONG  FIGURES. 


"  2  times  4  are  8  "  may  be  written  4X2  = 
8,  and  is  read  "4  multiplied  by  2  is  8," 
or  "2  times  4  equals  8." 

At  least  some  of  the  recitation  of  the 
above  table  should  be  as  follows :  "  2  times 
4  are  8,  2  in  8  four  times;"  "3  times  2 
arc  6,  3  in  6  twice,"  etc.  Always  recite 
these  tables  upward,  that  is,  2  times  4,  3 
times  2,  4  times  5,  etc. 

After  learning  this  table  test  the  pupils 
by  using  the  table  in  Teachers'  Edition,  p. 
63,  which  is  differently  arranged. 

1 .  John  bought  5  pencils  at  4  cents  each ; 
what  did  they  cost  ? 

Solution :  If  one  pencil  costs  4  cents, 
five  pencils  will  cost  5  times  4  cents,  or  20 
cents.  If  preferred,  take  the  following: 
They  will  cost  5  times  4  cents,  or  20 
cents. 

2.  What  cost  3  oranges  at  five  cents 
each  ? 

3.  If  one  sheet  of  paper  costs  2  cents, 
what  will  4  sheets  cost? 

4.  A  boy  bought  3  marbles  worth  2 
cents  each  ;  what  should  he  pay  for  them  ? 

5.  How  many  pints  in  5  quarts? 
Show  that  there  are  2  pints  in  a  quart 

by  pouring  2  pints  of  water,  one  at  a  time, 
into  a  quart  cup.     Show  in  a  similar  man- 
ner that  there  arc  four  quarts  in  a  gallon. 
See  Teachers*  Edition,  p.  71. 


»0  FIRST  STEPS  AMONG  FIGURES. 

Solution  :  In  one  quart  there  are  2  pints, 
in  fii'c  quarts  there  are  5  times  2  pints,  or 
10  pints. 

6.  How  many  quarts  in  3  gallons  ? 

7.  A  lady  bought  3  quarts  of  milk ;  she 
has  only  pint  tickets ;  how  many  tickets 
should  she  give  for  the  milk  ? 

8.  If  a  pig  should  be  fed  3  ears  of  corn 
at  one  feeding,  how  many  ought  he  to  have 
at  5  feedings  ? 

9.  Harry  has  5  cents  for  picking  one 
bushel  of  hops ;  how  many  cents  will  he 
have  for  picking  3  bushels  ? 

10.  A  boy's  mother  gave  him  3  cents 
for  each  armful  of  wood  he  brought  in  ;  he 
brought  in  4;  how  many  cents  did  he 
earn  ? 

11.  Three  pupils  have  each  3  books; 
how  many  books  have  they  altogether  ? 

12.  Jane  has  two  dolls;  Mary  has  3 
times  as  many;  how  many  has  Mary? 

Write  in  Arabic:  (13.)  CXI;  (14.)  XC  ; 
(15.)  CL;  (I6.)CXXV;  (I7.)XCVI;  (18.) 
CXIX  ;  (1 9.)  CLXXIV  ;  (20.)  LXXXVII ; 
(21..CLV;  (22.)XLVIII;  (23.)CI;  (24.) 
CLXI. 

Write  in  Roman  :  (25.)  ninety-six  ;  {26\ 
one  hundred  seventy-six ;  (27.)  one  hun- 
dred eight;  (28.)  ninety-nine;  (29.)  one 
hundred  forty-eight;  (30.)  one  hundred 
eighty-two. 


FIRST  STEPS  AMONG  FIGURES.  21 

Read  the  following  numbers:  (31.) 

25758;  (32)  74051;  (33)  16300;  (34.) 

81407;  (35.)  40520;  {36.)   60000;  {37.) 

70030;  (38.)  9010;  (39.)  18002;  (40.) 

27643- 

Write  in  Arabic : 

41.  Fifty  thousand  two  hundred  nine- 
teen. 

42.  Twenty-seven  thousand  thirty. 

43.  One  thousand  one  hundred. 

44.  Sixty-five   thousand    four  hundred 
ninety. 

45.  Seventy  thousand  twenty. 

46.  Forty-six  thousand  ninety-one. 

47.  Nine  thousand  eight. 

For  more  practice  see  Teachers'  Edition, 
page  69. 

48.  Count  by  4's  from  4  to  60. 

49.  Count  by  4's  from  I  to  17. 

50.  Add  23/33,  20,  3,  22,  13.  2,  33,  21, 
and  23. 

51.  Add  2,  33,  23,  31,  12,  3.  30,  23.  12, 
33.  and  32. 

52.  Add  33.  21.  13,  23.  3,  32,  23,  I,  23, 
30,  23,  and  33. 

53.  Add  23,  33,  30,  22,  13.  23,  30,  13, 
2,  32,  33.  and  20. 

54.  Add  3,  31,  23,  2,  33,  30,  23,  32,  22, 
32,  13,  and  33. 

See  Teachers*  Edition,  pp.  60,  61. 


22  FIRST  STEPS  AMONG  FIGURES. 

55.  Add    32.  21.    33,   3,  32.    20,   31,    23, 

13,  22,  31,  and  2. 

56.  Add  2.  33,  21,  S2,  13,  3,  23.  12,  32, 
33,  23,  30,  and  3. 

57.  Add  31,  12.  23,  33,  2,  23,  31,  33, 
2S,  22.  31,  and  S}. 

1.  In  a  school- room  there  are  5  desks 
in  each  row,  and  4  rows ;  how  many  desks 
in  the  room  ? 

2.  Lester  wasted  5  minutes  in  school 
ever)'  day ;  how  much  tii.ie  did  he  waste 
in  a  week  ? 

3.  Three  families  of  mice  live  in  the 
garret ;  there  are  five  mice  in  each  family  ; 
how  many  mice  in  the  garret  ? 

4.  How  many  quarts  in  5  gallons  ?  • 

5.  If  John  picks  5  quarts  of  cherries  one 
day  and  4  quarts  the  next,  how  many  does 
he  pick  in  the  two  days? 

6.  A  mother  who  had  5  hungry  boys 
made  2  loaves  of  bread  one  day,  3  the  next, 
and  4  the  third  day  ;  how  many  loaves  did 
she  make  in  the  3  days? 

7.  There  are  3  boats  on  the  lake,  and  4 
boys  in  each  boat ;  how  many  boys  on  the 
lake  ? 

8.  What  cost  3  balls,  if  one  ball  cost  3 
shillings  ? 

9.  'James  had  7  buttons  on  his  coat ;  he 
lost  off  3  ;  how  many  remained  on  ? 


HRST  STEPS  AMONG  FIGURES.  25 

10.  There  were  8  panes  of  glass  in  a 
window ;  a  boy  broke  2  of  them  with  his 
ball ;  how  many  whole  ones  were  there 
then? 

11.  A  man  paid  5  dollars  for  2  dogs; 
one  of  them  cost  3  dollars ;  what  did  the 
other  cost  ? 

12.  Katie  bought  3  yards  of  ribbon  at 
2  cents  a  yard  to  trim  her  doll's  bonnet ; 
how  much  did  the  trimming  cost  ? 

13.  If  one  top  costs  4  cents  what  will 
5  tops  cost  ? 

14.  I  exchanged  a  cord  of  wood  worth 
7  dollars  for  a  ton  of  coal  worth  five  dol- 
lars ;  how  much  did  I  lose  ? 

15.  Two  boys  went  nutting;  one 
brought  home  5  pecks  and  the  other  4 
pecks;  how  many  pecks  did  both  bring? 

16.  Frank  paid  4  shillings  for  a  pair  of 
doves  and  2  shillings  for  oats  to  feed  them  ; 
how  much  money  did  he  spend  for  them  ? 

17.  Emma,  Hattie,  and  Lucy  have  each. 
4  dolls ;  how  many  dolls  have  they  all  ? 

18.  Count  by  4's  from  i  to  61. 

19.  What  will  it  cost  to  ride  4  miles  on 
the  cars,  the  fare  being  2  cents  a  mile  ? 

20.  How  many  times  will  the  hands  of 
a  clock  go  from  XII  to  VI  in  4  days  ? 


^4  FIRST  STKPS  AMONG  FIGURES. 

D r V 1  >- 1 '  > v  T  \ ij LE.  j's  and  review. 

"  UJt  repeated,  long  remembered.** 

To  be  recited  ••  5  in  25  five  times,"  "2 
in  6  three  times."  etc. 


a 

b 

c 

25 

6 

10 

16 

3 

10 

9 

8 

12 

5 

2 

3 

5 
2 

4 

3 

2 

3 

4 

3 

5 

4 

I 

5 

3 

2 

4 

d 

e 

2 

15 

20 

6 

8  4 

2 

5 

4 

3 
2 

2  4 
4  I 

I 

3 

5 

/ 

ir 

20 

4 

IS 

] 

[2   5 

5 

2 

3 

4  5 

425  31 

"5  in  25  five  times"  may  be  written 
25-^-5  =  5,  and  is  read  "25  divided  by  5 
equals  5." 

I.  At  3  cents  each  how  many  lemons 
can  you  buy  for  1 2  cents  ? 

See  Teachers   Edition,  pp.  63,  73- 


FIRST  STEPS  AMONG  FIGURES.  25 

•  Solution :  If  one  lemon  costs  3  cents, 
for  12  cents  you  can  buy  as  many  lemons 
as  3  is  contained  times  in  12,  or  4.  If 
preferred,  use  the  following:  As  many  as 
there  are  3's  in  12,  or  4. 

2.  10  cents  will  buy  how  many  marbles 
at  2  cents  each  ? 

3.  Joseph  spent  12  cents  for  oranges, 
paying  4  cents  for  each  orange ;  how 
many  did  he  buy  ? 

4.  A  boy  sold  a  pair  of  rabbits  for  25 
cents ;  how  many  oranges  at  5  cents  each 
can  he  buy  with  the  money  ? 

5.  Mr.  Brown  paid  a  boy  12  shillings 
for  work,  at  the  rate  of  2  shillings  a  day ; 
how  many  days  had  the  boy  worked  ? 

6.  Lottie  spent  16  cents  for  candy  ;  she 
gave  4  cents  an  ounce ;  how  many  ounces 
did  she  buy? 

7.  At  3  cents  each  how  many  marbles 
can  Edward  buy  for  1 5  cents  ? 

8.  Mary's  brothers  gave  her  16  cents, 
each  giving  her  4  cents ;  how  many  broth- 
ers had  she  ? 

9.  Harry  has  8  dollars  in  the  bank  ;  his 
father  has  put  2  dollars  there  for  him  each 
birthday ;  how  many  birthdays  has  he 
Been  ? 

10.  If  one  pineapple  costs  2  shillings^ 
how  many  can  you  buy  for  10  shillings  ? 


26  '     FIRSI  STEPS  AMONG  FIGURES. 

1 1.  How  many  ink  wells  at  3  cents  each 
can  you  buy  for  9  cents  ? 

12.  If  the  fare  on  the  cars  is  4  cents  to 
a  certain  village,  how  much  is  the  fare  for 
both  ways? 

13.  Eight  little  girls  were  in  the  woods 
looking  for  violets ;  only  3  girls  found 
any  ;  how  many  found  none  ? 

14.  How  many  quarts  in  3  gallons? 

15.  How  many  pailfuls  of  beans  will  it 
take  to  fill  an  eight-quart  basket,  if  each 
pail  holds  2  quarts? 

16.  Fred  has  5  apples,  John  has  one, 
and  Harry  has  3  ;  how  many  have  all  ? 

Read  the  following  numbers:  (17.) 
321468;  (18.)  108320;  (19.)  716381; 
{20.)  400750;  (21.)  604025  ;  (22.)  700006  ; 
(23.)  800000;  (24.)  70016;  (,25.)  215000; 
{26.;  380500;  (27.)  50000. 

It  is  an  excellent  exercise  for  the  class 
to  write  the  foregoing  numbers  in  words. 

Write  in  Arabic : 

28.  Forty-nine  thousand  seven  hundred 
sixty. 

29.  Ten  thousand  ninety. 

30.  Three  hundred  seventeen  thousand 
nine  hundred  thirty-one. 

31.  Nine  hundred  thousand  one  hun- 
dred one. 

32.  Four  hundred  thousand  forty. 


FIRST  STEPS  AMONG  FIGURES.  27 

33.  Six  liundred  thousand. 

34.  Twt)  hundred  ninety-one  thousand 
rfive. 

35    Thirty  thousand  ten. 

36.  Add  32.  21.  13,  3,  32,  22,  12,  33, 
32,  13,  23,  and  2. 

37.  Add  3,  31.  23,  3,  33.  12,  2.  33,  21, 
32,  22,  33,  and  23. 

38.  Add  33.  21,  23,  31,  12,  3.  23,  13, 
32,  20,  32.  and  12. 

39.  Add  21,  33,  3,  12,  30,  23.  33.  ^^ 
J3,  33,  21,  and  33. 

40.  Add  32,  3,  23,  33,  13,  21.  23,  33, 
30,  20,  12,  2,  and  21. 

41.  Add  23,  33,  2,  30,  33,  13,  II.  32, 
^3,  33,  21.  and  32. 

42.  Add  43,  21,  4,  14,  32,  23,  42,  34, 
24.  42,  33.  and  4. 

43.  Add  4,  23,  42,  21,  3,  44,  43.  23,  31, 
14,  23.  and  31. 

1.  What  cost  3  books  at  4  shiUings 
•each '" 

2.  Mary  rode  in  the  swing  five  times, 
and  Jane  4  times ;  how  many  times  did 
they  both  ride  ? 

3.  How  many  marbles  at  two  cents  each 
can  you  buy  for  6  cents  ? 

4.  Jane  saw  five  doves  on  the  ground ; 
three  of  them  flew  away ;  how  many  re- 
mained on  the  ground? 

See  Teachers'  Edition,  p.  69. 


28  FIRST  STEPS  AMONG  FIGURES. 

5.  Bessie  went  to  school  5  days,  and 
Mary  went  3  ;  how  many  more  days  did 
Bessie  go  than  Mary  ? 

6.  How  many  times  can  I  take  2  mar- 
bles from  a  pile  of  8  marbles? 

7.  A  little  girl  had  20  cents ;  how  many 
four-ceni  lead  pencils  can  she  buy  with  her 
money  ? 

8.  A  boy  walked  2  miles  each  day  for 
4  days ;  how  far  did  he  walk  ? 

9.  A  boy  having  three  five-cent  pieces 
lost  two  of  them ;  how  many  cents  had  he 
left? 

10.  A  little  girl  ate  3  buckwheat  cakes 
for  breakfast,  and  her  brother  ate  4 ;  how 
many  did  both  eat  ? 

1 1 .  Lewis  gave  4  boys  4  marbles  apiece ; 
how  many  did  he  give  them  all  ? 

12.  Jennie  spent  5  cents  for  raisins,  3 
cents  for  candy,  and  2  cents  for  a  stick  of 
gum ;  how  much  did  she  spend  ? 

13.  On  the  east  side  of  a  house  there 
are  5  windows ;  3  of  them  are  open ;  how 
many  are  closed  ? 

14.  A  good  boy  carried  8  pails  of  water 
for  his  mother  on  Monday,  and  5  on  Tues- 
day ;  how  many  more  did  he  carry  on 
Monday  than  on  Tuesday? 

*  15.   How  many  quarts  in  4  gallons? 
16.   Among  how  many  children  can  I 


FIRST  STEPS  AMONG  FIGURES.  29 

divide  15  plums  that  each  may  receive  3 
plums? 

17.  There  are  16  towns  in  Ontario 
County ;  if  you  learn  the  names  of  4  of 
them  each  day,  in  how  many  days  will 
you  learn  the  names  of  all  of  them  ? 

1 8.  Charles  had  4  marbles  and  his  broth- 
er gave  him  5  ;  how  many  had  he  then  ? 

19.  Henry  had  to  stay  after  school  5 
minutes  for  whispering,  and  2  minutes  to 
solve  an  example ;  how  long  did  he  have 
to  stay  ? 

20.  If  John  gets  5  cents  for  husking  one 
bushel  of  corn,  how  many  bushels  must  he 
husk  to  earn  20  cents  ? 

21.  If  a  boy  traded  a  knife  worth  10 
cents  for  a  top  worth  5  cents,  how  much 
did  he  lose  ? 

22.  How  much  will  a  boy  earn  in  4 
days  at  3  shillings  a  day  ? 


1)D 

iTioN  Table. 

fs  and  review. 

6 
3 

a 

3  7      5 

4  5       6 

b 
2463 
7     3      4    5 

9    7     12     II  9    7     10     & 

See  Teachers'  Edition,  p.  74. 


30  FIRST  STEPS  AMONG  FIGURES. 

C  d 

7.524  7637 

6734  3567 


13      12      58  10      II      9      14 

^  / 

5246  3524 

3456  7456 


8     6     9     12  10     9     7     10 

g  h 

637  524 

7     3       4  567 


13     6     II  10     8     II 

Subtraction  Table.        fs  and  review, 

a  b 

13     12     5     8  10     II     9     7 

6734  3465 


524  7732 


c 


d 


14     9     7     12  II     9     7     10 

7     3     4       5  ^734 

7637  5246 


FIRST  STEPS  AMONG  FIGURES.  3 1 


e 

/ 

8 

9  10 

13 

6 

10 

6  10 

5 

4   7 

7 

3 

6 

4   5 

3 

5   3 

6 

3 

4 

2   5 

1?  9 

8 

II 

8 

II 

6  5 

3 

5 

6 

7 

645  624 

Multiplication  Table,  fs  and  review, 

a  b 

^375  2463 

3456  7345 


18 

12  35 
c 

30 

14 

12  24 
d 

15 

7 
6 

5  2 
7  3 

4 
4 

7 
3 

21 

6   3 
5   6 

30  18 

7 
7 

42 

35  6 

16 

49 

5 
3 

e 

2   4 
4   5 

6 
6 

3 
7 

/ 

5   2 
4   5 

4 
6 

15     8     20     36  21     20     10     24 


32  FIRST  STEPS  AMONG  FIGURES. 


6  3 

7  3 

7 
4 

5 
5 

h 

2      4 
6       7 

42     9 

28 

25 

[2      28 

Division  Table. 

7'^ 

r  tf «^  review. 

<x 

b 

49 
7 

i8     30 
6       5 

3       6 

21 

3 

7 

16 
4 

6     35 
3       7 

42 
6 

7 

4 

2       5 

7 

c 

d 

15 

3 

8     20 

4       5 

36 
6 

15 

5 

3 

24     12 
4       3 

6      4 

14 
7 

5 

2       4 

6 

2 

30 
6 

e 
35      12 

5       4 

18 
3 

10 

5 

/ 
20     21 

4       7 

24 
6 

5 

7       3 

6 

2 

5       3 

4 

42     9 
7     3 

28 
4 

25 

5 

12     28 
6       7 

637  52 

See  Teachers'  Edition,  p.  95. 


FIRST  STEPS  AMONG  FIGURES.  33 

To  the  teacher.  Read  carefully  the  pre- 
faces to  both  the  Pupils'  and  the  Teach- 
ers' Editions,  and  also  the  Special  Notice 
which  precedes  the  latter. 

Solve  the    following  examples    in    ad- 
dition : — 
I-  23     2  33     3.  31     4.  31     5.  32     6   21 
31         23         33  13         13  13 

33         31        22         12        33         32 
20         20        30        33         21  33 

13  13        23         30         12         20 


32 

33 

13 

22 

32 

32 

23 

32 

32 

31 

23 

3 

31 

23 

31 

13 

13 

31 

33 

31 

23 

23 

32 

31 

12 

33 

33 

31 

23 

23 

7. 

32 

8.33 

9-  31 

10.  32 

23 

31 

12 

23 

31 

23 

33 

31 

33 

33 

22 

23 

21 

12 

31 

13 

13 

23 

23 

31 

22 

31 

12 

20 

33 

32 

33 

33 

31 

23 

32 

23 

23 

33 

23 

31 

Some  of  these  examples  are  to  be  given 

"^^^^y-  See  Teachers'  Edition,  p.  88. 


34  FIRST  STEPS  AMONG  FIGURES. 

Solve   the   following    examples   in    ad- 
dition : — 

II.    32      12.  33      13.  23      14.  33     15.    32 


21 

21 

33 

22 

C3 

13 

32 

31 

31 

30 

33 

33 

22 

23 

33 

30 

13 

33 

20 

22 

2 

32 

21 

32 

31 

23 

3 

13 

31 

23 

31 

12 

21 

23 

33 

23 

30 

32 

33 

20 

32 

21 

33 

31 

3 

33 

33 

13 

23 

31 

23 

32 

33 

33 

33 

16. 

32 
23 
23 
30 
23 
31 
22 

32 
21 
32 
33 
23 

17. 

33 
22 

30 
21 

13 
32 

33 
22 

31 
23 
22 

31 

18. 

21 

33 
32 
23 
33 
22 

31 
23 
32 
13 
33 
31 

19- 

31 
23 
31 
23 
30 

33 
21 

13 
32 
32 
23 
32 

■   Some  of  these  examples  are  to  be  given 

daily.  See  Teachers'  Edition,  p.  89. 


FIRST  STEPS  AMONG  FIGURES.  35, 


Solve  the 

followin 

g  examp 

les  in  ud- 

dition : — 

20.  32 

21.  33 

22. 

31 

23.  23 

13 

21 

23 

31 

23 

33 

33 

23 

31 

32 

31 

33 

33 

22 

22 

30 

23 

23 

32 

23 

31 

31 

23 

12 

21 

32 

21 

33 

12 

13 

32 

23 

32 

21 

33 

32 

33 

23 

32 

33 

32 

33 

33 

23 

34  33 

25-  23 

26. 

31 

27.  13 

21 

31 

23 

33 

30 

22 

32 

21 

13 

30 

21 

32 

23 

3 

13 

33 

31 

32 

33 

23 

30 

52 

33 

33 

23 

23 

20 

20 

31 

33 

13 

12 

12 

31 

32 

33 

23 

32 

33 

32 

32 

33 

23 

12 

Some  of  these  examples  are  to  be  given 

daily. 


36  FIRST  STEPS  AMONG  FIGURES. 


Solve   the 

i   following 

examples 

in   ad- 

dition : — 

28.  32 

29-  33 

30 

.  21 

31.33 

20 

21 

32 

23 

33 

32 

23 

30 

23 

33 

33 

22 

12 

21 

21 

12 

33 

13 

23 

33 

31 

33 

33 

21 

22 

33 

13 

21 

33 

22 

23 

13 

32 

31 

33 

33 

21 

33 

21 

30 

33 

23 

32 

22 

32.  30 

33. 

21 

34. 

23 

23 

33 

32 

12 

13 

13 

31 

23 

23 

33 

32 

32 

22 

31 

33 

32 

32 

23 

23 

10 

32 

31 

II 

30 

33 

32 

31 

22 

23 

23 

31 

33 

22 

Some  of  these  examples  are  to  be  given 
daily. 


FIRST  STEPS  AMONG  FIGURES.  37 

After  these  examples  have  all  been  solved 
once,  unless  the  pupils  are  very  ready  in 
adding,  let  them  commence  at  the  first 
example  and  solve  them  all  again  It  is  of 
the  utmost  importance  that  the  pupil  should 
learn  to  add  quickly  and  correctly,  for  in 
practical  life  he  will  use  addition  a  dozen 
times  where  he  will  use  fractions  once. 

^Examples  in  Subtraction. 

35.  75986        36.  96897        37.  68795 
54234  72044  5250 


38.  69587        39.  95786        40.  68795 
47043  2053  6203 


41.  86975        42.  79586        43.  68579 
20324  2102  40234 


44,  76859        45.  97586        46.  68579 
4506  35430  5345 


•Examples  in  Multiplication. 
I.  23,103    2.   30,231    3.   13,202    4.  21,032 
2332 


See  Teachers'  Edition,  p.  91. 


3^  FIRST  STEPS  AMONG  FIGURES. 

5.  31,023  6.  12,320  7.  20,123  8.  23,132 

3233 


9.31.420    10.32,301     11.30,423    12.23,123 
2323 


•Examples  in  Division. 
13.  2)64.682      14.  3)93.609     15.  2)20,486 


16.  2)40,826 

17.   3)39.069 

18.  2)82,604 

19.   2)28,064 

ao.  3)90.639 

21.   2)26,048 

22.   2)80,264 

23.   3)60.936 

See  Teachers'  Edition,  p.  92. 


FIRST  STEPS  AMONG  FIGURES.  3^ 


•Examples  in 

Addition. 

24-  342 

25-  234 

26. 

434 

231 

423 

244 

423 

344 

342 

443 

244 

444 

334 

432 

322 

234 

343 

434 

422 

443 

324 

27.  213 

28.  444 

29 

.  431 

432 

231 

242 

444 

443 

344 

344 

232 

423 

231 

424 

344 

414 

344 

332 

342 

231 

421 

30.  34 

31.  243 

32. 

4 

442 

424 

342 

321 

331 

234 

434 

212 

23 

444 

443 

341 

343 

344 

422 

212 

424 

444 

243 

341 

341 

444 

234 

213 





•  The  teacher  U  to  ihow  the   pupils  how   to  solve   such 
cxan-.ples. 


40  FIRST  STEPS  AMONG  FIGURES. 


33.  134 

34.  324 

35. 

432 

423 

431 

341 

344 

344 

223 

103 

223 

312 

422 

431 

434 

344 

444 

^ 

244 

441 

324 

443 

343 

213 

332 

413 

444 

423 

36.  2342     37.  4323     38.  3423 


4324 

2433 

2342 

3432 

4344 

4214 

2343 

4224 

2343 

3423 

3433 

4242 

2343 

1234 

1234 

3434 

4322 

4321 

4232 

2443 

3M4 

1443 

3244 

3423 

3234 

4432 

4343 

4342 

2343 

2432 

See  Teachers*  Edition,  p.  93. 


FIRST  STEPS  AMONG  FIGURES.  4I 


39-  3424 

40.  4234 

4132 

3442 

3434 

2323 

1234 

4342 

4343 

1234 

2432 

3421 

1243 

4343 

3324 

2422 

4432 

3234 

3243 

4343 

4432 

2342 

41-  343        42.  341        43.  431 


231 

234 

343 

444 

443 

234 

332 

444 

321 

243 

321 

443 

444 

213 

444 

423 

442 

324 

411 

344 

242 

344 

431 

441 

432 

342 

304 

243 

214 

323 

444 

404 

444 

321 

341 

321 

42  FIRST  STEPS  AMONG  FIGURES. 


44-  213 

45.  341 

46.  423 

441 

232 

344 

334 

444 

231 

234 

323 

444 

443 

213 

324 

342 

441 

431 

134 

344 

344 

423 

422 

444 

444 

343 

321 

324 

243 

423 

443 

412 

344 

322 

333 

442 

203 

431 

322 



■ 

47'  3424 

48.  4234 

49.  4324 

4323 

3342 

3432 

3243 

4433 

4243 

4434 

1234 

2324 

2332 

3243 

4433 

3443 

4324 

3442 

4324 

2442 

2334 

2423 

3334 

4343 

4244 

4234 

2434 

3423 

2442 

2343 

3342 

3223 

4123 

4324 

4324 

3434 

3423 

3443 

4442 

2342 

4321 

3223 

See  Teacliers'  Edition,  p.  94. 


FIRST  STEPS  AMONG  FIGURES.  43 


50  3423 

51.  3443 

3244 

4234 

4334 

2343 

2433 

4324 

4242 

4432 

3423 

2444 

4344 

4321 

1234 

3443 

4422 

4242 

4342 

4433 

3423 

3444 

4444 

2344 

3444 

4323 

2331 

3434 

1.  Richard  has  7  cents  and  Oliver  has 
6  cents,  how  many  have  both  ? 

2.  Cora  bought  5  sticks  of  candy  and 
Hattie  4,  how  many  did  both  buy  ? 

3.  Herman  had  11  cents,  but  he  lost  3 
of  them,  how  many  had  he  then? 

4.  Herbert  had  7  marbles,  he  found  5 
more,  how  many  had  he  then  ? 

5.  Ella  had  8  new  needles,  she  broke  5 
of  them,  how  many  whole  ones  had  she 
then  ? 

6  Clara  solved  12  examples  while  Kitty 
solved  7,  how  many  more  did  Clara  solve 
than  Kitty  ? 

See  Teachers'  Edition,  p.  97. 


44  FIRST  STEPS  AMONG  FIGURES. 

7.  Anna  had  10  oranges,  she  gave  away 
4  of  them,  how  many  had  she  left  ? 

8.  Frank  had  4  pencils,  he  bought  3 
more  ;  how  many  had  he  then  ? 

9.  What  cost  7  lemons  at  4  cents  each  ? 

10.  If  an  orange  cost  5  cents,  what  will  3 
oranges  cost  ? 

11.  If  an  orange  cost  6  cents,  and  a 
lemon  cost  5  cents,  what  will  both  cost  ? 

12.  How  many  quarts  in  3  gallons  ? 

13.  Henry  had  13  apples,  he  gave  his 
brother  7  of  them  ;  how  many  had  he  then  ? 

14.  At  5  shillings  a  bushel,  what  will  6 
bushels  of  apples  cost  ? 

15  What  cost  7  pencils  at  3  cents  each  ? 

16  A  boy  walked  4  miles  in  the  morn- 
ing and  3  miles  in  the  afternoon.  How  far 
did  he  walk  ? 

17.  Samuel  has  7  cents  in  one  pocket 
and  5  cents  in  another,  how  many  in  both  ? 

18.  Carlos  had  9  cents  in  his  pocket,  he 
lost  4  of  them  through  a  hole  in  his  pocket, 
how  many  had  he  then  ? 

19  Mary  spelled  8  words  correctly  and 
Emma  spelled  6  words  correctly,  how  many 
more  did  Mary  spell  correctly  than  Emma? 

20.  At  3  cents  each,  how  many  marbles 
can  you  buy  for  15  cents? 

21.  12  boys  were  sliding  on  the  ice,  7  of 
them  fell,  how  many  remained  standing  ? 

Sec  Teachers'  Edition,  p.  98. 


FIRST  STEPS  AMONG  FIGURES.  45 


22.  If  a  boy  earn  7  cents  a  day,  how 
many  days  will  it  take  him  to  earn  42  cents  ? 

23.  If  one  lead-pencil  cost  6  cents,  how 
many  can  be  bought  for  30  cents } 

24.  James  had  12  cents,  he  spent  5  of 
them  and  lost  3  more,  how  many  had  he 
left  > 

Solution :  He  disposed  of  the  sum  of  5 
cents  and  3  cents  or  8  cents.  He  had  left 
the  difference  between  8  cents  and  12  cents 
or  4  cents;  or,  if  he  spent  5  cents  and  lost 
3,.  he  disposed  of  the  sum  of  5  cents  and 
3  cents  or  8  cents.  If  he  had  12  cents  and 
disposed  of  8  cents  he  had  left  the  difference 
between  12  cents  and  8  cents  or  4  cents. 

25.  Sarah  had  10  needles;  she  gave  3 
of  them  to  Nellie  and  4  of  them  to  Martha. 
How  many  had  Sarah  left  ? 

26  Herbert  had  6  cents,  he .  earned  7 
cents  and  spent  5  cents  ;  how  many  had  he 
then  > 

27.  Joel  had  7  cents,  he  earned  5  cents 
and  found  4  cents  :  how  many  had  he  then  ? 

28  How  many  days  will  it  take  Walter 
to  walk  28  miles,  if  he  walks  4  miles  each 
day  } 

29.  An  orange  cost  6  cents  and  a  cocoa- 
nut  3  times  as  many.  How  much  did  the 
cocoanut  cost? 


46  URST  STEPS  AMONG  FIGURES. 

30.  Mary  had  11  cents,  she  lost  5,  and 
earned  enough  to  make  her  number  9. 
How  many  did  she  earn  ? 

31.  There  are  4  columns  in  John's  spell- 
ing lesson  and  5  words  in  each  column. 
How  many  words  in  his  lesson  ? 

32.  Some  boys  are  out  flying  kites,  the 
wind  blows  down  two  kites  and  7  less  3  re- 
main.    How  many  at  first  in  the  air  ? 

33.  A  boy  hoed  corn  for  4  cents  a  row 
and  earned  24  cents.  How  many  rows 
did  he  hoe  ? 

34.  A  man  can  walk  a  mile  in  10  minutes, 
he  starts  from  his  home  and  walks  to  town 
in  5  minutes.  How  far  from  town  does  he 
live  > 

35.  I  had  10  cents.  I  bought  2  two-cent 
stamps,  and  gave  4  cents  to  a  poor  little 
boy  for  bread  ;  how  many  left  for  candy  ? 

36  There  arc  9  ten  o'clock  scholars  this 
morning  and  one  more  than  one  third  of 
them  left  their  books  at  home.  How  many 
of  them  brought  their  books  ^ 

37.  Willie  had  13  cents  and  he  spent  6 
of  them  tor  candy  ;  how  many  had  he  left  ? 

38  How  many  sponges  at  6  cents  each 
may  be  bought  for  36  cents  ? 

39.  What  cost  a  pencil  at  7  cents  and  a 
pint  of  peanuts  at  5  cents  ? 


FIRST  STEPS  AMONG  FIGURES.  47 

40.  What  cost  7  pencils  at  5  cents  each  ? 
Count  by  5's  from  2  to  62. 
Review  counting  by  5's  from  I  and  $  to 
61  and  60.     Review  counting  by  4*3. 
Count  by  5's  from  3  to  63. 
Count  by  5!s  from  4  to  64. 


345 

2.  35 

3.  405 

452 

454 

352 

553 

542 

544 

245 

135 

255 

523 

523 

533 

345 

454 

441 

521 

542 

355 

453 

5 

523 

45 

352 

344 

35^ 

534 

555 

424 

435 

434 

345 

5-  505 

6.  305 

532 

453 

543 

454 

344 

444 

325 

535 

354 

543 

413 

532 

355 

553 

445 

324 

435 

354 

513 

214 

543 

454 

543 

455 

545 

345 

544 

353 

553 

353 

48  FIRST  STEPS  AMONG  FIGURES. 

Examples  in  Subtraction. 

7.  7<^,6^7  8.  96.487 

64.252  31.343 


9.  79.689  10.  687,985 

43.264  34,532 


II.  795.869  12.  47.685 

43.543  5.232 


13.  96.758  14.  5.879    • 

54.428  2,343 

15.  764.358        16.  764.037        17.  423.703 

201,223  23,010                12,40c 


18.  647,094  19.  7A>^^7 

24.070  3,053 


20.  74,687  21.  695,047 

30.435  252,004 


22.  52^,012       23.  613.021        24  431,024 
46,566  56.554  55.357 


See  Teachers'  Edition,  p.  lOI. 


FIRST  STEPS  AMONG  FIGURES.  49 

25.    634,210        26.    820,132         27.    431,201 
56,643  53/657  54464 


28     942,031         29.    731,203         30.    841,310 
66,466  64,636  63,653 


31.  312,403   32.  420,314   33-  341.531 
45.257       54.652       34.354 


34.  213,042  35.  314.253  36.  453.621 
36,527     51.646     32,365 


37.  324,102  38.  231,430  39.  425.301 
46,637     12,257     51,625 


40.  430,221   41.  320,413   42.  534.102 
52,147      63,257      62,637 


43.  534,210   44-  130,241   45-  342,013 
16,436      53.725      30,157 


46.  703.524   47-  423.102   48.  624,130 
26,352      47.056      61,563 


50  FIRST  STEPS  AMONG  FIGURES. 

If  more  practice  is  desired  at  this  stage 
the  foregoing  examples  may  be  reviewed. 
Count  by  6's  from  6  to  60. 

I.  4532  2.  3541  3.  5423 


3215 

4325 

4352 

5453 

5432 

3544 

3344 

3254 

5435 

2535 

5545 

4334 

4253 

3251 

2453 

5432 

4314 

3545 

2354 

2443 

4334 

4235 

5425 

5442 

5542  4543  2355 


M  3524 

5.  4325 

4352 

3552 

3445 

5445 

5334 

3453 

4523 

4524 

5452 

5335 

3445 

3452 

4334 

5344 

5523 

2325 

4354 

4555 

See  Teachers'  Edition, 

p.  103. 

FIRST  STEPS  AMONG  FIGURES.  5r 


6.  3542 

7-     245 

8.  5342 

4354 

4532 

3534 

2435 

2454 

4253 

5043 

5321 

5425 

3530 

3^43 

343 

354 

4254 

4534 

4245 

5435 

2345 

5432 

2353 

5234 

3554 

540 

4553 

4322 

3254 

3425 

2453 

4342 

5342 

5245 

5535 

2435 

9-  3045 

10.  3452 

4532 

5544 

5450 

2^45 

3345 

4532 

2534 

5453 

350 

13H 

4003 

5045 

5435 

3453 

3542 

4234 

4354 

4553 

5425 

5345 

4543 

3522 

52  FIRST  STEPS  AMONG  FIGURES. 

Multiply: 

II.    2,314  12.   4,231  13.    2,014 

2  2  2 


14.  30,124  15.  21,302 

2  3 


16.  34,201  17.  23,041 

2  2 


The  teacher  should  show  the  pupils  how 
to  solve  the  following  examples : 

18.  35,246  19.  42,356 

2  3 


20.  46,352  31.  35.642 

2  3 


22.  36,452        23.   26,453       24.  46,352 
3  3  2 


25.  64,526      26.  53,624      27.  64,524 
3  4  3 


See  Teachers'  Edition,  p.  104. 


FIRST  STEPS  AMONG  FIGURES.  53 

38.  53,426   29.  64,536   30.  35>642 


31.    46,352  32.    36,426         33.    35.246 

3  4  3 


34  36.452      35-  36,546      36.  64,526 
6  6  6 


37-  46,352      38-  63,524      39.  35,042 
5  4  4 


40.  40,536      41-  30.462      42.  46,035 
5  6  5 


43.  26.304    44.  25,036    45.  52,064 
6  6  6 


Count  by  6's  from  I  to  6 1. 
Count  by  6's  from  2  to  62. 
Count  by  6's  from  3  to  63. 
Count  by  6's  from  4  to  64. 
Count  by  6's  from  5  to  65. 


54  FIRST  STEPS  AMONG  FIGURES. 


Addition  Table.              S's  and  review. 

a 

b                               C 

74852 
34    5    ^7 

63    74    8     5    263    7 
8345678345 

10  8  13  II  9 

14  6  II  9  14   12  10  9  7  12 

d 

37485 
56783 

/ 
2637485263 
4567834567 

8  13  II  16  8 

6  II  9  14  12   II  9  7  12  10 

g 
74    8    5 
83    4    5 

//                    i 
2       6*348       526 
6       7867       834 

15  7  12  10 

8     13  II  10  15      13  5  10 

Subtraction  Table.      8's  and  revienr. 

a 

b                     c 

10  12  7  15  8 
5    43    86 

13  II  9  7  12    10  II  12  14 
7    845    67387 

58472 

63526      3847 

d 
9  II  6  8  10    ] 
65434 

/ 
14  5  13  8  13    II  9  14  II  16 
63856      77848 

36256 

82537      42678 

g 
96  12  13  8 

5  3    7    5  4 

h                     i 
10  10  9  7     12  10  15  II 
3834       5676 

43    5    84 

7263       7485 

See  Teachers'  Edition,  p.  105. 

FIRST  STEPS  AMONG  FIGURES.  SS 


Multiplication  Table. 

(5"j 

and 

'  review. 

/? 

^ 

c 

7 

4 

8 

5 

2 

63    7    4 

8 

5 

2    6 

3 

3 

4 
i6, 

5    6 
4030 

_7 
14 

83    4    5 

6 

7 

35 

8    3 
16  18 

4 

21 

48  9  28  20 

48 

12 

./ 

e 

/ 

7 

3 

7 

4 

8 

52    6    3 

7 

4 

8    5 

2 

5 

5    6 
1542; 

7    8 
2864 

34    5    6 
15  830  18 

7 
49 

8 

3    4 

5 

35 

12  24  20 

10 

^ 

h 

i 

6 

3 

7 

4 

8 

5    2    6 

3    4 

.     8 

52 

0 

6 

_2_ 

8 

3 

_4 

5    6    7 

8    6 

:  _z 

83 

_4 

3621  56  12  32  2512422424  5640624 


28948  1430 
43    8    7    6 

b                         C 
40  16  21  2048    35  16  18  12 
54356      7834 

73625 

84748      5263 

35245642  15 
56765 

/ 
24640  28  64    15  8  21  36 

43    8    7    8      34    7    <^ 

74873 

62548      5236 

g 
1020243249 

54387 

h                     i 
18303225  12   422456  12 
65456     7883 

25847      36852     6374 

See  Teachers'  Edition,  p.  105. 

56  FIRST  STFPS  AMONG  FIGURES. 

1.  George  paid  15  cents  for  a  knife,  and 
after  breaking  it  sold  it  for  8  cents.  How 
many  cents  did  he  lose  ? 

2.  Lewis  bought  a  reader  for  6  shillings 
and  an  arithmetic  for  4  shillings  ;  how  much 
money  did  he  spend  for  both  ? 

3.  Anna  is  13  years  old  and  Mary  is 
5  years  old.  How  many  years  older  is 
Anna  than  Maiy  } 

4.  John  had  8  cents ;  he  lost  5  of  them, 
how  many  had  he  left  ? 

5.  What  cost  4  books  at  8  shillings  each  ? 

6.  How  many  oranges  at  5  cents  each 
can  you  buy  for  40  cents  .? 

7.  Walter  spent  16  cents  for  pears  at  2 
cents  each  ;  how  many  pears  did  he  get  ? 

8.  Jane  had  15  needles;  she  lost  3  of 
them  and  broke  5  ;  how  many  had  she  left  ? 

9.  Byron  rode  down  hill  4  times  one 
afternoon  and  his  father  twice  as  many 
times.  How  many  times  did  Byron's  father 
ride  down  hill  ? 

10  William  had  26  cents  ;  he  lost  two  of 
them  and  spent  the  rest  for  marbles  at  3 
cents  each.  How  many  marbles  did  he 
buy  ^ 

1 1 .  Charles  was  sent  to  the  store  to  buy 
6  spools  of  thread  at  8  cents  a  spool.  He 
took  50  cents  with  him,  how  much  change 
should  he  take  home  ? 

See  Teachers'  Edition,  p.  io8. 


FIRST  STEPS  AMONG  FIGURES.  57 

12.  A  boy  having  8  cents  earned  5  cents 
and  his  sister  gave  him  2  cents,  how  many 
had  he  then  } 

13.  A  boy  set  2  traps  in  the  woods ;  2 
rabbits  went  into  one  trap  and  twice  as 
many  went  into  another.  How  many  went 
into  both  ? 

14  A  gun  carriage  has  four  wheels  ;how 
many  wheels  have  7  gun  carriages  ^ 

15.  If  it  takes  6  horses  to  draw  one  can- 
non, how  many  horses  will  draw  8  cannons  } 

16  There  are  a  sergeant  and  6  privates 
at  one  picket  post,  and  a  corporal  and  4 
privates  at  another.  How  many  soldiers  at 
both  > 

17.  If  3  oranges  cost  12  cents  what  will 
one  orange  cost  ? 

Solution  :  If  ///r^^  oranges  cost  I2  cents^ 
<?«^  orange  will  cost  one-third  of  12  cents 
or  4  cents 

We  get  J  of  a  number  by  dividing  it  by 
3,  i  of  a  number  by  dividing  it  by  4,  etc. 

What  is  one-fourth  of  24 } 

What  is  one-third  of  21  .? 

What  is  one  si.xih  of  42  ?  One-fifth  of 
30  ^  One-fourth  of  32  ?  One-seventh  of 
56?     One-half  of  14?     One-sixth   of    18. > 

18.  If  8  horses  eat  24  bushels  of  oats  in 
a  week,  how  many  bushels  will  one  horse 
eat? 


58  FIRST  STEPS  AMONG  FIGURES. 

10.  If  42  cents  is  the  price  of  7  marbles, 
what  is  the  price  of  one  marble  ? 

20.  4  boys  have  28  cents,  and  each  have 
an  equal  number.  How  many  cents  has 
each  boy } 

21.  A  farmer  has  18  pigs  in  3  pens  and 
the  same  number  in  each  pen.  How  many 
pigs  in  one  of  the  pens  ? 

22  If  8  pears  cost  16  cents,  what  cost  i 
•pear  ? 

23.  At  4  cents  each,  how  many  lemons 
can  be  bought  for  28  cents  ? 

24.  If  three  tops  cost  15  cents,  what 
cost  I  top? 

25.  What  cost  I  bushel  of  oats,  if  7 
bushels  cost  35  shillings  ? 

26.  12  dollars  will  buy  how  many  birds 
at  2  dollars  each  ? 

2y.  If  5  hens  cost  20  shillings  what  will 
I  hen  cost  ? 

28.  How  many  peaches  at  2  cents  each 
can  you  buy  for  16  cents  ? 


FIRST  STEPS  AMONG  FIGURES.  59 


1.  5304       2. 

453 

3-  5243 

4243 

304 

4524 

3524 

5035 

3452 

5452 

4540 

5335 

535 

3235 

3543 

4343 

2424 

4254 

3425 

5353 

5435 

5254 

4545 

3542 

2542 

3424 

5354 

4335 

4353 

4435 

3454 

2532 

3423 

5543 

4245 

5354 

4325 

5454 

3543 

4  4305 

5-  3544 

3450 

5434 

5234 

4355 

2543 

3243 

5455 

5424 

3213 

4552 

4543 

2345 

2355 

5433 

5432 

2253 

4343 

4534 

2524 

3445 

3455 

2343 

3532 

4554 

See  Teachers*  Edition  p.  no 


6o  FIRST  STEPS  AMONG  FIGURES. 


6.  4305            7 

2345 

8.  5435 

3453 

5334 

554 

435 

4243 

3443 

5044 

3545 

5545 

3453 

5434 

4353 

4545 

3552 

5534 

5432 

3345 

3444 

3354 

5453 

5554 

3545 

4245 

2345 

34 

3524 

4235 

5432 

4305 

5453 

4355 

5353 

3544 

4544 

3445 

5355 

9.  3452 

10.  3452 

2345 

5435 

5443 

4544 

4534 

3353 

4455 

«  5445 

5243 

3234 

3524 

5544 

4355 

4353 

5533 

2455 

4432 

4434 

5454 

5554 

3345 

5432 

5432 

3245 

FIRST  STEPS  AMONG  FIGURES.  6t 

Count  by  6*s  from  2,  3,  4  and  5,  to  62, 
63,  64  and  65. 

Read  the  following  numbers : 
I.  750000748  2.  90000047 
3.  680000740  4.  700746000 
5.  12003000  6.  750908716 
7.  801000071  8.  679374819 
9.  7 1 50 1 6390  10.  900060000 
II.  7000000  12.  800000005 
Addition  : 

Finding  how  many  units  there  are  in  two 
or  more  numbers  and  expressing  them  in 
one  number  is  called  addition. 

The  number  found  by  addition  is  called 
the  sii7n  or  amoimt. 

Proof  Add  the  columns  both  upward 
and  downward,  and  if  the  results  agree  they 
are  probably  correct. 

11.  Write  in  Arabic  fifteen  mil.  ten  th. 
ninety. 

12.  Write  in  Arabic  three  hun.  fifty  mil. 
nine  hun.  th. 

13.  Write  in  Arabic  eight  mil.  five  hun. 
forty-three  th.  seven. 

14.  Write  in  Arabic  one  hun.  mil.  six  hun. 
thirty-two. 

15.  Write  in  Arabic  seventy-five  mil. 
three  hun   th. 


62  FIRST  STEPS  AMONG  FIGURES. 

i6.  Write  in  Arabic  fifty  mil  fifty  th.  fifty. 

17.  Write  in  Arabic  two  hun.  mil.  sixty 
th.  four. 

18.  Write  in  Arabic  one  hun.  five   mil. 
one  hun   five. 

19.  Write  in  Arabic  one  hun.  nineteen 
mil  forty  th. 

20.  Write  in  Arabic  three  hun.  eight  mil. 
thirteen  th.  two  hun.  eighty-one. 

21.  Write  in  Arabic  twenty  th. 

22.  Write  in  Arabic  fifty  mil.  fifty. 

23.  Write  in  Arabic  three  th.  two  hun. 
forty-five. 

24.  Write  in  Arabic  nineteen  mil.  five  hun. 
th. 

25.  Write   in    Arabic    eight    mil.  three 
hun. 

26.  Write  in  Arabic  four  hun.  mil. 

27.  Write  in  Arabic  seven  hun.  sixty-one 
mil.  five  hun.  sixteen  th.  twenty. 

28.  Write  in  Arabic  six  hun   th. 

29.  Write  in  Arabic  fifty  th.  forty. 

30.  Write  in  Arabic  three  hun.  eight. 

31.  Write  in  Arabic  nine  th. 

32.  Write.in  Arabic  six  hun.  mil.  five.  hun. 

33.  Write  in  Arabic  sixty  mile,  four  hun. 
th.  three  hun. 

34.  Write  in  Arabic  three  mil.  fifteen  th. 
thirty. 

See  Teachers'  Edition,  p.  114. 


FIRST  STEPS  AMONG  FIGURES.  6^ 

•35.   Write  in  words  I3,(X)4,020. 

36.  Write  in  words  300,216,000. 

37.  Write  in  words  232,341,519. 

38.  Write  in  Roman  fourhun.  sixty-three. 

39.  Write  in  Roman  eight  hun.  forty-four. 

40.  Write  in  Arabic  DCLXXVII. 

4 1 .  Write  in  Roman  two  hun.  eighty-nine. 

42.  Write  in  Arabic  XCVIII. 

43.  Write  in  Arabic  CDXI. 

44.  Write  in  Roman  seven  hun.  nineteen. 

Subtraction : 

Taking  one  number  from  another  num- 
ber is  called  subtraction. 

Remainder  or  difference. 

The  number  found  by  taking  one  num- 
ber from  another  is  called  the  difference  or 
remainder. 

In  subtraction  the  number  to  be  sub- 
tracted is  called  the  subtra/iend,  and  the 
number  it  is  subtracted  from  is  called  the 
minuend.  The  result  in  subtraction  is 
called  the  difference  or  remainder. 

Proof.  Add  the  remainder  and  the  sub- 
trahend, and  if  the  result  equals  the  min- 
uend the  work  is  probably  correct. 

The  pupils  should  often  be  required  to 

•  Notice  that  such  compound  words  as  seventy-five^ 
forty-one,  sixty-nine,  etc,  require  a  hyphen. 


64  FIRST  STEPS  AMONG  FIGURES. 


write  the  name 
in  subtraction, 

of  each  number 
as  follows : 

in  examples 

321  Minuend. 
45  Subtrahend. 

276  Remainder 

or  di£ 

Subtraction. 

(O 

(2.) 

(3) 

8»342.053 
474.377 

73.520,031 
243.075 

81,400,253 
2,302,675 

(4) 

(5.) 

(6.) 

71,420,035 
241,068 

93,520,014 
5,700,043 

4.531.024 
765.337 

(?•)  . 

(8.) 

(9.) 

74,200,352 
5,402,568 

6,314.025 

565,068 

73.500.241 
212,376 

(10.) 

(II.) 

(12.) 

94,002,531 
7,006,763 

81,350,024 
576,546 

34,200,156 
7,620,478 

See  Teachers'  Edition,  p.  u8. 


FIRST  STEPS  AMONG  FIGURES.  65 


(13)  (14.)  (15) 

7»352,o34    95,300,421    863,005,241 
181,376      420,376     2,307,374 


(16.)        (I7-)        (18.) 
93,510,042    35,200,416     7.300,425 
46,350,357     2,540,348      602,557 


(19.)        (20.)        (21.) 
^3,001,425    95.320,041    85,241,300 
5,005,743     4460,474     7.534.720 


(22.) 
94,300,521—570,376=? 

(23.) 
42,530.014—6.765,068=? 

(24) 
7i»300,524— 1,000,677=? 

(25) 
82,400,153—4.740,075=? 


66  FIRST  STEPS  AMONG  FIGURES. 

The  teacher  will  choose  between  the  two 
following  sets  of  definitions  : 

Multiplication. 

A  short  method  of  adding  equal  numbers 
is  called  muliiplication. 

Multiplicand. 

One  of  the  equal  numbers  is  called  the 
viultiplicand. 

Multiplier. 

The  number  which  shows  how  many  of 
the  equal  numbers  are  used  in  adding  is 
called  the  multiplier. 

In  the  example  542  -»-  542  +  542,  542  is  the 
multiplicand  and  3  isihe  multiplier,  and  we 
solve  the  example  as  follows  : 

542  Multiplicand, 
3  Multiplier, 


1,626  Product. 
Or  the  following  definitions  may  be  used: 

Multiplication. 

Taking  a  number   a   certain    number  of 
times  is  called  multiplication. 

Multiplicand. 

The    number   taken,    (or    multiplied,)  is 
called  the  multiplicand. 


FIRST  STEPS  AMONG  FIGURES.  6/ 

Multiplier. 

The  number  by  which  we  multiply,  (or 
which  shows  how  many  times  the  multipli- 
cand is  taken,)  is  called  the  multiplier. 

Product. 

The  result  of  the  multiplication  is  called 
\^^  product. 

Proof. 

I.  Multiply  the  multiplier  by  the  multi- 
plicand, and  if  the  product  equals  the  pro- 
duct first  obtained,  the  work  is  probably  cor- 
rect. 

II.  Or  divide  the  product  by  the  multi- 
plicand, and  if  the  work  is  correct  the  re- 
sult will  be  the  multiplier,  or  divide  by  the 
multiplier  and  get  the  multiplicand. 


Multiplication. 
I.     35,246       2.     42.356 
7                      7 

3  35.642 
6 

4-     57.463 
7 

5.     35.642 
7 

6.  35.246 
8 

7.     57.463 
8 

8.     35.246 
6 

9.  46,058 
8 

See  Teachers'  Edition,  p.  119. 


68  FIRST  STEPS  AMONG  FIGURES. 

10.    685,037       II.    485,067      12.    857,046 
%  7  6 


13.   637,058  14.    364.078 

4  7 


15.  3.640,758^8=.?       16.  7.583.640x7=? 

When  there  is  more  than  one  figure  in  the 
number  by  which  we  multiply,  the  right 
hand  figure  of  the  result  is  placed  under 
the  figure  we  multiply  by. 


17.  46.352        18.  56,342 

19-  2.635 

32               23 

32 

20.    364,526x43  =  ? 

21.   35.264x45=? 

22.  4^563>^54=? 

23.  534,652x46=? 

24.  645.362x64=? 

25.  63.527x76=? 

26.  4675x67=? 

27.  57.463x74=? 

FIRST  STEPS  AMONG  FIGURES.  69 

28.  357»426x63=? 
29-  47063x67=? 

30.  63,527x54=? 

31.  352,746x34=? 

32.  475.063x57=? 

33.  630,574x76=? 

34.  57.463x35=? 

35.  357426x47=? 

Division. 

Finding  how  many  times  one  number  is 
contained  in  another  is  called  division. 

Dividend. 

The  number  which  contains  the  other  is 
called  the  dividend.  (If  preferred,  the 
number  divided  is  called  the  dividend.) 

Divisor. 

The  number  which  is  contained  in  the 
other  is  called  the  divisor.  (If  preferred, 
the  number  by  which  we  divide  is  called 
the  divisor?) 

Quotient. 

The  number  found  by  dividing  is  called 
the  quotient. 


70  FIRST  STEPS  AMONG  FIGURES. 

Proof. 

Multiply  the  quotient  by  the  divisor, 
and  add  the  remainder  if  there  be  one ;  if 
the  work  is  correct  the  result  should  equal 
the  dividend. 

Divisor,  7)4321  Dividend. 

Quotient,     617 — 2  Remainder. 

Short  Division. 

(I.)  (2.)  (3.) 

2)6,846  3)612,921  4)3,281,224 


(4.)  (5.)  (6.) 

3)9,182,715        4^2.082,836        5)5.304.535 


(7.)  (8.)  (9) 

3)24,152,721    2)128,166414   4)836,284,820 


(10.)  (II.)  (12.) 

3)21,186,912       4)2,032,828     2)106,416,818 

See  Teachers'  Edition,  p.  120. 


FIRST  STEPS  AMONG    FIGURES.  J I 

With  remainders.  ♦ 

13-  7.372-3  =  ? 

14.  14.173^4=? 

15.  16,427-^3=? 

16.  823,975-3  =  ? 

17.  2,182,939-^4=? 

18.  1,327.178-^-5=? 

19.  954.546^4=? 

20.  5•663,032-^3=? 

21.  26,173,385-^4=? 

22.  273.823-7-5=? 

23.  925,588-7-6=? 

24.  257.4294-4=? 

25.  3,741,254^6=? 
26   3/6.233--5=? 

27.  3.804.168-7-7=? 

28.  4.873.2904-5=? 

29-  4.497.190^7=? 

30.  3.880.9164-6=? 

31.  3,146.1534-4=? 

32.  172.4484-7=? 

33.  20.740,9654-6=? 

Count  by  7*s  from  7  to  42. 

•  Teach  the  division  series  with  remainders  in  Teaciv 
era'  Edition,  p.  107,  before  solving  these  examples. 


7^ 


Hksl  slLI'b  AMONG  FIGURES. 


Addition  Table.  g's  and  review. 

a  b  c 

695847369584736 

4    5    6    7    8.  94    5    6    7|  8    9    45    6 

101411  15  121167  II  13  121613  11812 

d  e  f 

9584    7|3    695    8|  27369 

7    8    9    4    5i6    7    3    9    41  96789 

i6  13  17    8  1 2!9  13  17  14  1 2!  1 1  13  10  14  18 


5    8    4 
4    5    6 


7    3 
7    8 


9  13  10  14  II  15 


9 

5 

8 

4 

7 

3 

4 

5 

6 

7 

8 

9 

13 

10 

14 

II 

15 

12 

Subtraction  Table.        p'.f  and  review, 
a  be 

13913  10 


12  15  II  14  II 
98768 


3    7 


4    8    3 
d 
14  18  9  12  14 
89549, 


15  13  10  14 
9  4  5  7 

10 
6 

r  6  9  5  7 

4 

5467 


85    7    3 

/ 
7  8  12  9  1317  13  16  12  8 
8456    7!  98765 


6    94    8    5    94    73 


II  13  16  12  15  II 
498765 
748596 


6'  8    5    963 
h 
7  16  10  14  II  15  12 
4945678 

3769584 


See  Teachers'  Edition,  p.  123. 


FIRST  STEPS  AMONG  FIGURES. 


75 


Multiplication  Table,  gs  and  review. 


69584 
45678 


73695. 
94567 


c 

8473 
8945 

63  12305435  !64  36  28  15 

e  f 

8275 
4967 


2445305632 

d 
69584!  73695 

6789    4I  56789 

36  61  40  72  1635  18  42  72  45  32  18  42  21 

695847316958473 
89456789456789 

48  81  20  40  24  49  24  154  36  25  48  28  56  27 


(/s  and  review. 


Division  Table. 

a  b  c 

32  563045  2463  12  305435  1643628  18 


87654 


48596 


94567 


73695 


5 

d  e 

42  1835  1672I4063  367245 
76549;  87689 
6    3    7    4    81  5    9    6    9    5 


8949 


8 

4    7 
/ 

2 

32 

2042 

21 

4 

5    6 

7 

8 

4    7 

3 

// 

48  81  49  2440  18  5/ 

89765   29 

69748    96 


362448  28  5627  25 

4867895 


9384735 


Sec  Teachers'  Edition,  p.  124. 


74  FIRST  STEPS  AMONG  FIGURES. 

1.  In  a  school  there  are  8  recitation 
rooms,  each  room  has  7  benches ;  how 
many  benches  in  these  rooms  ? 

2.  There  were  60  freight  cars  on  a  road 
and  6  of  them  were  destroyed  in  a  collision  ; 
how  many  u  mained  ? 

3.  If  a  man  can  draw  7  loads  of  sand 
in  I  day,  how  many  days  will  it  take  him  to 
-draw  42  loads  ? 

4  In  a  class-room  there  are  6  seats,  and 
€ach  seat  will  hold  8  pupils  ;  how  many 
pupils  can  be  seated  in  the  room  ? 

5.  Six  sheep  were  put  into  a  flock  con- 
taining Sy  ;  how  many  then  in  the  flock  ? 

6.  Pineapples  are  6  cents  each  ;  how 
many  can  be  bought  for  48  cents  ? 

Count  by  7's  from  7  to  70. 

7.  If  Nellie  pays  5  cents  for  candy  and 
has  8  cents  left,  how  many  cents  had  she 
at  first  ? 

8.  How  many  legs  have  7  cats  ? 

9    How  many  quarts  in  8  gallons  ? 

10.  How  many  quarts  in  12  pints  .^ 

11.  If  John  buys  some  candy  for  18 
cents,  some  peanuts  for  7  cents  and  an 
orange  for  6  cents,  how  many  cents  does 
he  spend  ? 

See  Teachers'  Edition,  p.  123. 


FIRST  STEPS  AMONG  FIGURES.  75 

12  *  If  8  sheep  cost  $32,  what  will  one 
sheep  cost  ? 

13  How  many  fingers  and  thumbs  have 
^  boys  ? 

14.  Willie  bought  2  pounds  of  crackers 
at  8  cents  a  pound,  and  half  a  pound  of 
cheese  at  14  cents  a  pound;  how  much 
money  did  he  spend  ? 

15.  Harry  had  a  ten-cent  piece,  a  five- 
cent  piece  and  two  three-cent  pieces  ;  how 
much  money  had  he  ? 

16.  Annie  had  60  cents  ;  she  spent  30 
cents  for  a  doll,  and  received  10  cents  and 
2  more  from  her  mother  for  doing  an 
errand  ;  how  many  had  she  then  ? 

17.  Thomas  had  13  chickens  and  11 
little  turkeys ;  the  cat  caught  5  of  the 
chickens  and  the  rats  caught  4  of  the  little 
turkeys  ;   how  many  of  both  were  left  ? 

Count  by  y's  from  i  and  2  to  71  and  72. 

18.  William  had  7  cents  and  John  had 
twice  as  many  ;  how  many  had  both  the 
boys  ? 

19.  I  have  18  pupils  in  a  spelling  class; 
4  of  them  misspell  some  of  their  words  ; 
how  many  recite  perfectly  ? 

*The  character  $    means  dollars  and  $31  U  read  thirty- 
two  dollars. 


y6  FIRST  STEPS  AMONG  FIGURES. 


20.  George 

had  7  glass  agates,  8  china 

and  6  common 

marbles  ;  how  many  marbles 

did  he  have  ? 

21.  If  24  apples  be  equally  divided  among 

8  boys,  how  many  will  each  have  ? 

22.  Walter  took  56  cents  to  the  store  to 

buy  sugar  at  8 

cents  a  pound  ; 

how  many 

pounds  could  he  get  ? 

23.  Katie's  i 

Tiother  gave  her  9 

cents,  her 

father  7  cents 

,  and  her  aunt  4 

cents ;  she 

bought  3  oranges  at  4  cents  each  and  spent 

the  rest  of  the 

:  money  for  candy 

at  2  cents 

a  stick  ;   how 

many   sticks  of 

candy  did 

she  get  ? 

I.  5435 

2.  3454         3. 

5342 

4354 

5323 

3455 

3343 

2345 

5534 

5425 

4534 

2345 

2342 

5453 

4453 

3454 

3225 

5225 

5535 

4544 

3542 

4344 

2345 

4354 

5432 

5432 

5335 

2345 

4543 

4543 

4554 

1234 

3254 

3435 

5432 

5432 

5342 

3555 

1234 

4554 

4343 

4555 

3343 

5443 

5321 

See  Teachers'  Edition,  p.  126. 


FIRST  STEPS  AMONG  FIGURES. 


4.  3452 

5. 

4532 

5324 

3254 

4543 

4325 

5435 

5543 

3354 

3454 

5445 

3434 

4554 

2345 

3425 

5432 

5432 

4545 

2345 

3454 

4553 

5235 

5434 

4523 

4423 

5445 

3545 

4354 

5434 

5432 

6.  4356    7 

.  3564 

8.  6345 

3645 

6453 

2436 

5234 

2566 

6563 

6563 

4635 

3656 

3656 

5364 

5365 

4345 

6532 

6534 

5456 

3656 

4655 

6563 

4364 

5362 

3625 

5635 

6656 

5356 

6556 

3545 

6545 

3446 

6666 

78  FIRST  STEPS  AMONG  FIGURES. 


9-  3456 

JO.  6354  . 

6563 

566s 

5635 

6536 

6366 

3456 

3556 

636s 

6645 

6653 

4536 

5546 

5663 

3665 

6636 

6236 

6355 

6463 

5663 

3554 

11.  684,632-47,757=?  Ans.  636,875. 

12.  43,126-2,765  =  .^     Ans.  40,361. 

13-  935.643-77.387=?  Ans.  858,256. 

14.  754,231-26,154=?  Ans.  728,077. 

15.  836,425-68,268  =  ?  Ans.  768,157. 

16.  364.135-71.543  =  ?  Ans.  292,592. 

17.  463.845-98,536=?  Ans.  365.309. 

18.  763,843-46,452  =  ?  Ans.  717,391- 


FIRST  STEPS  AMONG  FIGURES.  79 


19.  6453     20, 

•  3456 

21.  3645 

3566 

6566 

6556 

4635 

5635 

2663 

6326 

4364 

6356 

5663 

5662 

5565 

4566 

6556 

4636 

3656 

6636 

6353 

6365 

3665 

5664 

5636 

5326 

6656 

6656 

6553 

3533 

5625 

3466 

6665 

5564 

6536 

3456 

4563 

5462 

6363 

22.  5663 

23 

6543 

6656 

2345 

3546 

6234 

6635 

5623 

5366 

4562 

4653 

3456 

6636 

6566 

3565 

5335 

6366 

4666 

5436 

5656 

6563 

6535 

3666 

6666 

6525 

3553 

See  Teachers'  Edition,  p.  126. 


8o  FIRST  STEPS  AMONG  FIGURES. 

When  there  is  a  cipher  or  ciphers  in  the 
multiplier  between  significant*  figures,  do 
not  use  it  in  multiplying,  since  nothing 
times  any  number  is  nothing  ;  but  be  care- 
ful to  write  the  first  figure  of  each  product 
under  the  figure  you  multiply  by. 

1.  57.364x304=? 

2.  47*563  X  504=? 

3.  346,752x706=? 

4.  630,475  X  607=? 

5.  50.746x467=? 

6.  46.375x564-"=? 

7.  3.526x736=? 

8.  6.375x657=? 

9.  68,574x87=? 

10.  47,586x68=? 

11.  647,583x85=? 

12.  364.758x48=? 

^  13    105,743-4=? 

14.  3,176.207-5=? 

15.  1,418,497-4=? 

16.  22,477,527-^6=? 

17.  2,629,369^4=? 

18.  23,176,234-^5=? 

♦The  significant  figures  are  i,  2,  3}  4*  5,  6,  7,  8  and  9. 
See  Teachers'  Edition,  p.  128. 


FIRST  STEPS  AMONG  FIGURES.  8l 

19  30,249,7424-4  =  ? 

20  2,^25,160-^6  =  ? 

21.  31.760,850    7-? 

22.  34,459.582      6  =? 

23.  34,574,849     63? 

When  the  divisor  is  not  contained  in  the 
partial  dividend,  write  a  cipher  in  the  quo- 
tient and  the  partial  dividend  will  be  the 
remainder  to  be  prefixed  to  the  next  figure 
of  the  dividend.  The  teacher  will  illustrate 
by  the  following  examples  : 

24.  10,421^4  =  ? 

25.  44,593.672^7=t 

26.  2,115,885-^5=? 
27-  37.548,510-^-6=? 

28.  25,233,295-^7=? 

29.  37,229,621^8=? 
^30.  42,519,206-^-6=? 

31.  40,379,014-7=? 

32.  52,584.387^8=? 

33.  20,326,941-^8=? 

1.  874,895-^201=? 

2.  728,556^201=? 

3.  12.756.034-^-3,012=-? 

4.  8,604,338-7-2,031=? 
4* 


82  FIRST  STEPS  AMONG  FIGURES. 

5.  16.154,855-- 3,024  =  ? 

6.  49,486,701-^2,032=? 

7.  2,595,960^4,023  =  ? 

8.  1,460,703-^403  =  ? 

9.  10.357,440-3,024=? 

10.  14,244,539-=- 3,014--=? 

11.  13,822,604^4,034-? 

12.  25,079,486-7,036=? 

13.  38.968.336-^6.035=? 

14.  275.881,300-5,046=? 
15-  334.654.184-70,364=? 

16.  349,143,867-^6,057=? 

17.  226,396,593-60,453=? 

18.  18,651,776-4,076=? 

19.  526,026,567^7,046=? 

20.  236,326,931-5.074=? 

21.  39.361.095^6,024=? 

22.  540.611,445-7,068=? 

23.  450,299. 1 32--6o,378=? 

24.  3  2  7.040,029  -  5 ,048 = ? 

25.  413.535496-70.485  =  ? 

26.  241,993,12-5,086=? 

27.  546,695,551-^80,574=? 

28.  465,406.942^7,068=? 

29.  289,561,188^6,075  =  ? 

See  Teachers'  Edition,  p.  136. 


FIRST  STEPS  AMONG  FIGURES.  8^ 


30. 4563  31 

.  5465 

32.  4536 

3656 

6536 

5665 

6366 

6653 

4354 

5665 

3566 

6635 

4656 

6632 

5666 

6546 

5366 

6563 

6665 

6556 

6356 

5656 

3465 

3665 

3456 

6634 

5636 

6543 

5663 

6565 

5655 

2556 

3663 

6366 

6665 

6356 

5621 

5,326 

6565 

33. 5634 

34 

•  3456 

6565 

6536 

3456 

5665 

6563 

3566 

4656 

6343 

5665 

4656 

6363 

3565 

5636 

6666 

3565 

5356 

6366 

4663 

5636 

6536 

6653 

5465 

3546 

6323 

FIRST  STEPS  AMONG  FIGtJRES. 


1 .  A  boy  had  1 5  marbles  and  lost  all 
but  six  of  them ;  how  many  did  he  lose  ? 

2.  Mary  comes  to  school  5  days  in  a 
week ;  how  many  days  does  she  come  in 
8  weeks  ? 

3.  There  are  55  sticks  of  candy  in  a 
jar;  if  8  little  girls  each  buy  a  stick,  how 
many  sticks  will  be  left  in  the  jar  ? 

4.  How  many  marbles  can  a  boy  buy 
for  27  cents  at  three  cents  apiece  ? 

5.  Fanny  had  8  cents  and  Julia  had  9 
cents,  how  many  did  both  girls  have  ? 

When  an  example  involves  several  opera- 
tions the  pupil  should  give  but  one  at  a 
time. 

6.  How  much  more  will  6  oranges  cost 
at  4  cents  each,  than  7  peaches  at  2  cents 
each  ? 

Solution  :  If  one  orange  cost  4  cents,  6 
oranges  will  cost  6  times  4  cents  or  24 
cents.  If  one  peach  cost  2  cents,  7  peaches 
will  cost  7  times  2  cents  or  14  cents.  If 
the  oranges  cost  24  cents  and  the  peaches 
14  cents,  the  oranges  cost  as  much  more 
than  the  peaches  as  the  difference  between 
24  cents  and  14  cents  or  10  cents. 

7.  Henry  had  25  cents; he  gave  3  cents 
each  to  his  brother  and  sister,  spent  5  cents 
for  an  orange  and  2  for  candy  ;  how  many. 
cents  had  he  left  ? 

See  Teachers'  Edition,  p.  137. 


FIRST  STEPS  AMONG  FIGURES.  85 

8.  Harvey  had  a  twenty-five  cent  piece, 
a  ten-cent  piece,  a  five-cent  piece  and  a 
three-cent  piece ;  how  much  money  had  he? 

9.  If  I  had  4  apples  and  found  as  many 
more,  and  ate  two  of  them,  what  part  of  a 
dozen  had  I  then  ?  What  are  they  worth 
at  12  cents  a  dozen  ? 

10  I  have  a  clock  that  strikes  every 
quarter  hour  ;  how  many  times  will  it  strike 
in  9  hours  ? 

11  William  spent  12  cents,  James  spent 
one  third  as  many  and  three  cents  more; 
how  many  did  James  spend  ? 

12.  There  are  2  little  dogs  passing  ;  how 
many  eyes,  ears  and  feet  have  they  ? 

13.  Three  men  each  take  three  bags  of 
wheat  to  mill,  and  each  bag  contained  2 
bushels ;  how  many  bushels  did  the  men 
take  to  the  mill  ? 

14.  When  milk  is  6  cents  a  quart,  how 
many  quarts  can  you  get  for  42  cents  ? 

15.  When  milk  is  4  cents  a  quart,  how 
many  pints  can  you  get  for  20  cents  ? 

16.  In  a  school-room  there  are  7  rows  of 
seats,  and  6  seats  in  each  row  ;  how  many 
seats  are  there  in  the  room  ? 

17.  A  lady  made  7  squares  of  patch- 
work, and  her  little  girl  sewed  so  many  that 
one-half  of  what  both  sewed  was  10 ;  how 
many  did  the  little  girl  sew  ? 


86  FIRST  STEPS  AMONG  FIGURES. 

1 8.  How  many  boxes  of  wafers  at  6  cents 
a  box  may  be  bought  for  9  sheets  of  paper 
at  2  cents  a  sheet  ? 

19.  How  many  barrels  of  apples  at  $3  a 
barrel  can  be  given  for  6  yards  of  flannel  at 
32  a  yard  ? 

20.  How  many  four-horse  teams  can  be 
arranged  from  20  horses  ? 

21.  Three  fields  have  each  3  trees,  under 
each  tree  are  3  cows;  how  many  cows  in: 
the  three  fields  ? 

22.  A  man  bought  a  duck  at  9  cents  a 
pound  and  paid  54  cents  for  it ;  how  much- 
did  the  duck  weigh  } 

23.  If  6  oranges  cost  24  cents,  what  cost 
8  oranges } 

Solution  :  If  6  oranges  cost  24  cents,  ond 
orange  will  cost  one-sixth  of  24  cents,  or 
4  cents,  and  8  oranges  will  cost  8  times  4 
cents,  or  32  cents. 

24.  If  a  boy  walks  15  miles  in  3  days^ 
at  the  same  rate,  how  far  will  he  walk  in  4 
days? 

25.  If  it  takes  16  yards  of  cloth  for  2 
suits  of  clothes,  how  many  yards  will  it 
take  for  6  suits  ? 

26.  If  a  boy  goes  8  feet  in  stepping  4 
times,  how  far  will  he  go  in  stepping  7 
times? 

See  Teachers'  Edition,  p.  138. 


FIRST  STEPS  AMONG    FIGURES.  87 

27.  If  3  men  can  cut  9  acres  of  grain  ii> 
one  day,  how  many  acres  ca  1  6  men  cut  in 
a  (lay  ? 

28  If  it  takes  12  buttons  for  3  vests, 
how  many  buttons  will  it  take  for  8  vests? 

29  How  many  yards  of  cloth  at  $3  a 
yard  can  be  bought  lor  4  barrels  of  flour  at 
$6  a  barrel  ? 

30.  If  3  men  can  build  .a  wall  in  6  days^ 
how  long  will  it  take  one  man  ? 

31.  If  5  men  can  mow  a  field  of  grass 
in  10  days,  how  long  will  it  take  one  man  ? 

32.  If  4  men  cut  8  cords  of  wood  in  a 
day,  how  many  cords  will  i  man  cut  in  a 
day  ? 

33.  If  3  men  cut  a  pile  of  wood  in  9  days, 
how  long  will  it  take  one  man  ? 

34  If  3  mowing  machines  will  cut  27 
acres  of  grass  in  one  day,  how  many  acres 
will  7  mowing  machines  cut  in  one  day? 

35.  If  a  boy  earn  63  cents  in  7  days,, 
how  much  will  he  earn  in  6  days  ? 

36.  How  many  books  at  4  shillings  each 
can  you  buy  for  8  dozen  eggs  at  2  shillings 
a  dozen  ? 

37.  A  teamster  drew  8  loads  of  stone 
each  day  for  7  days ;  how  many  loads  did 
he  draw  ? 

38.  A  boy  gained  7  cents  by  selling  a 
knife  for  42  cents  ;  what  did  it  cost  him  ? 


88  FIRST  STEPS  AMONG  FIGURES. 

39.  William  worked  8  hours  at  2  shillings 
an  hour,  and  Henry  worked  3  hours  at  3 
shillings  an  hour  ;  how  much  did  both  earn  ? 

40.  Four  girls  have  each  2  hens,  and 
each  hen  has  6  chickens :  how  many 
chickens  have  the  four  girls  ? 

Read  the  following  numbers: 

1.  750406300. 

2.  4576000043 

3.  860000307. 

4.  1 5000045001. 

5.  7845678437. 

6.  37 1 474 1 5006. 

7.  47583000000. 

8.  370015300 

9.  40C00036700. 
10.  71000100000. 

Write  in  Arabic  the  following  numbers  : 

1.  Fifteen  n^l.  ten  th   three 

2.  Two  hun   eight  bil.  one  hun.  th. 

3.  Three  bil   twenty  mil.  six. 

4.  Thirteen    bil    nine    th.    seven    hun. 
forty-five.  ^ 

5.  Ninety-one  bil.  one  mil.  one  th.  one. 

6.  Four  bil.  seven  hun.  fifteen. 

7.  Two  hun.  sixty  mil. 

8.  One  bil.    three   hun.  sixty  mil.  two 
hun.  th. 

See  Teachers'  Edition,  p.  140. 


FIRST  STEPS  AMONG  FIGURES.  89 

9.  Five  mil.  ninety. 

10.  Write  in  Roman  nine  hun.  thirty- 
four. 

1 1.  Write  in  Roman  seven  hun.  forty-six. 

12.  W^rite  in  Arabic  DCCCXCVII. 

13.  Write  in  words  709460371000. 

14.  Write  in  Arabic  ten  th.  thirty. 

Teach  the  pupils  that  the  figure  at  the 
right  expresses  units  of  the  first  order,  the 
ne.xt  figure  to  the  left,  units  of  the  second 
order,  the  next  figure,  units  of  the  third 
order,  and  so  on. 

15.  Write  7  units  of  the  5th  order,  4  of 
the  3d  and  i  of  the  ist  (in  one  number.) 

16.  Write  3  units  of  the  8th  order,  5  of 
the  7rh,  9  of  the  3d  and  4  of  the  2d. 

It  may  aid  the  pupil  in  solving  the  follow- 
ing examples,  to  put  small  numbers  in  the 
place  of  the  ones  given,  and  see  how  it 
would  be  solved  without  the  slate,  then 
solve  in  the  same  manner. 

Simple  problems  for  the  slate,  involving 
Addition,  Subtraction  and  Multiplication  : 

1.  Mr.  Rogers  had  746  bushels  of  wheat 
and  sold  197  bushels  of  it;  how  much  had 
he  left  ? 

2.  George  had  295  cents  and  his  father 
gave  him  75  more  ;  how  many  had  he  then  } 


^O  FIRST  STEPS  AMONG  FIGURES. 


3.  Mr.  Smith  had  96  bushels  of  oats 
and  Mr.  Jones  had  9  times  as  much  ;  how 
many  bushels  had  Mr.  Jones } 

4.  Lewis  has  Sy  marbles  and  John  has 
just  as  many  ;   how  many  have  both  boys? 

5  Mr.  Howard  drew  8  loads  of  oats  to 
market,  and  there  were  79  bushels  in  each 
load  ;  how  many  bushels  did  he  draw  to 
market } 

6  From  the  sum  of  79  and  268,  take  158. 

7.  How  much  will  a  teacher's  salary 
amount  to  in  14  years,  at  $875  a  year.-* 

8.  James  has  47  marbles  less  than  John, 
and  John  has   174;  how  many  has  James? 

9  John  lost  15  cents  by  selling  his 
knife  for  90  cents  ;  what  did  it  cost  ? 

10.  Miles  took  2341  steps  in  going  to 
school,  and  Marcus  took  560  ;  how  many 
more  steps  did  Miles  take  than  Marcus  ? 

11.  Mr.  Decker  borrowed  $150  and  paid 
-$6$  of  it ;  how  much  does  he  still  owe  ? 

12.  What  will  46  bushels  of  barley  cost 
at  167  cents  a  bushel  ? 

13.  A  clerk  received  a  salary  last  year  of 
^ICXX).  He  spent  $260  for  board  and  $378 
for  clothing  and  other  expenses  ;  how  much 
money  did  he  save  ? 

14.  There  are  30  days  in  June,  and  31 
each  in  July  and  August ;  how  many  days 
in  these  three  summer  months  ? 

See  Teachers'  Edition,  p.  143. 


FIRST  STEPS  AMONG  FIGURES.  9 1 

15.  There  are  i68  acres  in  Mr  Fox's 
farm,  and  j\Ir.  Norton's  farm  contains  89 
acres  more  than  Mr.  Fox's ;  how  many 
acres  in  both  larms  ? 

16.  A  man  put  $950  in  the  bank  ;  he 
drew  out  $78  at  one  time,  $45  at  another, 
and  $159  at  another;  how  much  had  he 
left  in  the  bank  ? 

17.  John  had  39  marbles  and  Ezra  had  13 
more  than  twice  as  many  ;  how  many  had 
Ezra  ? 

18.  Mr.  Brown  bought  a  farm  for  $8460 
and  sold  it  for  $10380;  how  much  did  he 
gain  ? 

19.  What  cost  369  bushels  of  oats  at  68 
cents  a  bushel  ? 

20.  I  have  216  bushels  of  potatoes  in  3 
bins  ;  there  are  59  bushels  in  one  bin  ant 
98  bushels  in  another;  how  many  bushels 
in  the  third  bin  } 

2 1.  A  boy  having  85  cents,  bought  a  top  for 
1 8  cents,  a  ball  for  2  5  cents,  and  some  oranges 
for  27  cents;  how  many  cents  had  he  left? 

22.  If  there  are  y6  bushels  of  corn  in  a 
bin  that  will  hold  950  bushels,  how  many 
more  bushels  of  corn  may  be  put  into  it  ? 

23.  A  farmer  filled  at  one  time  29  bags 
with  oats,  and  at  another  47  bags.  If  he 
put  two  bushels  in  each  bag,  how  many 
bushels  of  oats  were  put  in  all  the  bags  ? 


92  FIRST  STEPS  AMONG  FIGURES. 


24.  From  one 

mUlion 

eight  hundred 

thousand  take  fifteen  thousand  ninety. 

I.  4567    2 

.  7645 

3.  4756 

7654 

5774 

3567 

6347 

6547 

6635 

7576 

5732 

7456 

4757 

7665 

6573 

5674 

4576 

5746 

7565 

5766 

7455 

4757 

7537 

6567 

5664 

6452 

7736 

7575 

3776 

6564 

2346 

6565 

6475 

4.  7546 

5 

.  7654 

4757 

6537 

5675 

5465 

6757 

7756 

7577 

4675 

4664 

3456 

7357 

6747 

5675 

7564 

4567 

► 

6475 

7654 

7746 

4735 

4653 

See  Teachers'  Edition,  p.  144. 


FIRST  STEPS  AMONG  FIGURES.  93 


6.  3456 

7-  6754 

8.  5746 

7653 

5673 

7457 

6575 

7566 

6575 

5747 

4375 

7^7 

7356 

7647 

5734 

5676 

5774 

4757 

4747 

6757 

7676 

5635 

7577 

AS^7 

4567 

4652 

^7}>S 

7476 

5767 

S^SJ 

5745 

77^6 

AS7^ 

6574 

5473 

7A^S 

9  4576 

10.  6457 

7654 

7564 

5773 

5735 

7566 

4657 

6757 

7576 

5674 

4657 

7365 

3456 

4537 

7573 

3456 

5746 

6765 

7757 

7S7^ 

4567 

4457 

3456 

94  FIRST  STEPS  AMONG  FIGURES. 

11.  4,362,516-754,359=? 

12.  736,952  —  78,672? 

13.  642,534-26,356=? 

14.  352,432-86,354=  ? 

15.  6,425,314-374.321=? 

16.  463,524-71,876=? 

17.  425,362-17,654=? 

18.  364,253-86,174=? 

19.  463,521-186,357? 

20.  483,654-91,987=? 

21.  635,245-12,567.? 

22.  837,524-259,286=? 

23.  43,452,431-4.238,865=? 

24.  756,324-85.543-=? 

25.  4.738,536-973.659=? 

26.  86,357-7,269=.? 

27.  34.023-9,876=.? 

28.  45.300,435-2.430,526=? 

29.  74, 200,03  2  —  5,1 40,05  4  =  ? 

30.  43,250,001-5,726,025=? 

31.  3,400,564-210,739=? 

32.  45.700,325-8.730,153=? 

1.  796,845  <89=? 

2.  479,685  x98=? 


FIRST  STEPS  AMONG  FIGURES.  95 


3.  68,975x79=? 

4.  647.583x467=? 

5.  68.574x456=? 

6.  486,075  X  807=  ? 

7.  58,697x64=? 

8.  9.687x75=  ? 

9-  4.796  X  39=  ? 

10.  85,974x84=? 

1 1.  79,685  X  69=  ? 

12.  68.974x79=? 

13.  49,786x85=? 

14.  59,068  X  604=  ? 

15.  70,968x907=  ? 

16.  70,309  X  709=  ? 

17.  860,479x709=? 

18.  759.068  X407  ? 

19.  47,096x609=  ? 

20.  748,609  X  507=  ? 

21.  58,709x608=  ? 

22.  96,047  X  709=  ? 

23.  97,806x597=? 

24.  79.689x4,759=? 

25.  896,748x6,978=  ? 

26.  7,580,065-1,251,298=? 

See  Teachers'  Edition,  p.  145. 


96  FIRST  STEPS  AMONG  FIGURES. 

2T.  3,740,683-923.754=? 

28.  735»035 -26,326=  ? 

29.  7430.246- 7,503,472=? 
I.  3,241,402-^7=? 

2.    5,678,005-7-8  =  ? 

3.  3,802,457^8=? 

J^  49,167,544^9=? 

5.  5.076.335-9=? 

6.  3»372,o8i-^9=? 
7-  3.725.891-8=? 

8.  52,301,166-^8=? 

9.  48,094,605^7=? 

10   43.719.125-9=? 

11.  52,629,186^8=? 

12.  40,224,713-7=? 
15.  67,217,191-9=? 

14.  608.226,845-9=? 

15.  37.100.695-^8  =  ? 

16.  460.241.323^7=? 

17.  4,781,158,859-^6=? 

18.  4.885,157.761-7=? 
19-  389,158,560-^6=? 

20.  5,240.869-7=? 

21.  3.356.977-7=? 

See  Teachers'  Edition,  p.  14& 


FIRST  STEPS  AMONG  FIGURES.  97 

22.  6.727741-8=:? 

23.  5,115,271-8  =  ? 

24.  21,367,398-^5,024=? 

25.  3,722,901^-607  =  ? 

26.  14,204,241-403  =  ? 

27.  158,632,783-30,135=? 

28.  38.693,395-6,024  =  ? 

29.  38,464,365^50.396  =  ? 

30.  31.970,764-^7.048  =  ? 

31.  3,891,687.541^60,475=? 

32.  2,925.490,533^60,479=? 

33.  462,857,740^8.069=  ? 

34.  2,934.401,497-^70,586=? 

35.  226,663,766^60.379=  ? 

36.  194,513.933^40.297=? 

37.  1 84, 1 1 0,903 -^  5,048=  ? 

38.  390.761,546-6,037=  ? 

39.  32.688,027,778-5-70,486=? 

40.  674,476,820-^9.037=  ? 

41.  27,971,095-6.074=? 

42.  355,2I2.265-^5,036=? 
43-  23.377.796-5.024=  ? 
44  452,113,508-^7,056=? 
45.  299.019,935-^406=? 


98  FIRST  STEPS  AMONG  FIGURES. 

46.  2,408,592,665  ^  5,064=  ? 

47.  36,979.544-^8,026=? 

48.  426,696,721-7,014=? 

49-  437.791.333-6.038=? 

50.  216,526,004-^5,027=  ? 

51.  2.464,085,695-^604,978? 

52.  238,049.090-^ 5;037=  ? 

53.  4,110,929.380^70.586=? 

54.  283,899.778^6,034=? 

55.  4,980,403.784^-70,496=  ? 

56.  2,202,155-^463=? 

57.  251,864^361=? 

58.  434,801-573  =  ? 

59.  268,500^463=? 

60.  463,7994-582=? 
61.*  356,116-365=? 

62.  357,243^465=? 

63.  403,123^586=? 

64.  4i4,6i7--473  =  ? 

65.  4,174,696^485=? 

•  When  the  left  hand  figure  of  the  divisor  is  equal  to 
the  left  liand  figure  of  the  dividend,  if  the  next  figure  of 
the  divisor  be  greater  than  the  next  figure  of  the  di\a- 
dend,  point  off  as  if  the  left  hand  figure  of  the  divisor 
were  greater.  The  divisor  (in  one  step  of  the  opera- 
tion) never  is  contained  more  than  nine  times. 
See  Teachers'  Edition,  p.  148. 


FIRST  STEPS  AMONG  FIGURES.  99^ 


^'    5499.513^796  =  ? 

67.  5.538,824^684  =  ? 

68.  7,084.249^896=? 
69    30.734,480^645=? 

70.  43,722,966-573=? 

71.  270,578,240^-4,035  =  ? 

72.  268,439,581^4,657-? 

1.  William  paid  54  cents  for  6  doves  ; 
what  did  each  dove  cost  ? 

2.  In  an  orchard  there  are  6  rows  of  trees 
and  7  trees  in  each  row  ;  how  many  trees 
in  the  orchard  ? 

3  Henry  has  8  cents  in  one  pocket  and 
9  cents  in  the  other  ;  how  many  cents  has 
he? 

4.  James  has  8  apples  and  his  sister  has 
6  ;  how  many  more  has  James  than  his  sis- 
ter ? 

5.  What  cost  9  knives  at  7  shillings  each  ? 

6.  A  boy  paid  25  cents  for  a  ball  and  sold 
it  for  18  cents  ;  how  many  cents  did  he  lose  ? 

7.  If  I  pencil  cost  4  cents,  what  will  8 
pencils  cost  ? 

8.  George  bought  a  knife  for  8  shillings, 
aball  for  5  shillings,  and  a  bat  for  2  shil- 
h'ngs  ;  what  did  he  pay  for  all  ? 

9.  Marcus  spent  8  cents  for  lemons  at  4 
cents  each  ;  how  many  lemons  did  he  buy  ? 

See  Teachers'  Edition,  p.  150. 


lOO  FIRST  STEPS  AMONG  FIGURES. 

lo  If  5  men  cut  ten  cords  of  wood  in  a 
day,  how  many  cords  will  7  men  cut  ? 

11.  If  2  men  can  dig  a  certain  ditch  in  4 
days,  how  long  will  it  take  one  man  to  dig 
it? 

12.  If  4  men  can  cradle  12  acres  of  grain 
in  one  day,  how  many  acres  will  one  man 
cradle  in  a  day  ? 

1.3.  How  many  rods  of  wall  will  one  man 
build  in  a  day,  if  3  men  build  9  rods  in  one 
day  ' 

14.  li  3  boys  can  pick  the  stones  from  a 
meadow  in  9  days,  how  many  days  will  it 
take  one  boy  to  pick  them  ? 

15.  How  many  weeks  in  35  days? 

16.  If  4  pounds  of  sugar  cost  36  cents, 
what  cost  8  pounds  ? 

1 7.  When  a  pineapple  costs  18  cents  and 
an  orange  costs  6  cents,  how  much  more 
does  the  pineapple  cost  than  the  orange  ? 

18.  If  a  boy  can  walk  12  miles  in  4 
hours,  how  far  can  he  walk  in  5  hours  ? 

19.  If  4  men  can  do  a  piece  of  work  in 
8  days,  how  long  will  it  take  one  man  ? 

30.  Jane  bought  5  figs  for  3  cents  each, 
and  a  yard  of  cloth  for  9  cents ;  how  much 
did -she  pay  for  all? 

See  Teachers'  Edition,  p.  151 


FIRST  STEPS  AMONG  FIGURES.  lOI 

21.  Mary  sews  4  hours  each  day,  how 
many  hours  does  she  sew  in  a  week  ? 

22.  How  much  will  a  man's  board  for  a 
week  cost  at  4  shiUings  a  day  ? 

23.  How  much  will  a  man  earn  in  a 
week,  if  he  gets  9  shillings  for  a  day's  work  ? 

24.  If  15  cats  are  on  a  wall  and  every 
third  cat  jumps  off,  how  many  are  left  ? 

25.  There  are  8  quarts  in  a  peck,  how 
many  pecks  in  32  quarts  ? 

26.  How  many  quarts  in  3  pecks  ^ 

27.  A  boy  picked  16  quarts  of  beans  and 
sold  them  at  25  cents  a  peck  ;  how  much 
money  should  he  receive  .•* 

28.  Charles  has  7  cents  and  his  brother 
3  more  than  twice  as  many  ;  how  many 
have  both  ? 

29  On  Monday  morning  Mary  had  20 
sticks  of  candy  ;  she  ate  2  each  day,  how 
many  had  she  left  the  next  Monday  night  ? 

30.  Arthur  had  1 1  peaches,  he  ate  3 
and  gave  his  sister  half  of  the  rest ;  how 
many  did  he  keep  ^ 

31.  How  many  marbles,  2  for  4  cents, 
can  you  get  for  18  cents? 


I02  FIRST  STEPS  AMONG  FIGURES. 

EXAMPLES  FOR  THE  SLATE. 
If  the  pupil  will  use  small  numbers  in* 
stead  of  the  large  ones  in  tlie  following 
examples,  and  think  carefully  how  he  would 
solve  them  if  they  were  not  for  the  slate, 
and  then  do  the  same  with  the  numbers 
given,  using  the  slate  as  a  help,  he  will  be 
greatly  assisted. 

1.  A  man  had  $3,210,  he  spent  $978  for 
wheat  and  $74^  for  corn  ;  how  much  money 
had  he  left  ? 

2.  What  is  the  product  of  9,687  and  75  ? 
3*  From  the  sum  of  3796  and  4279,  take 

their  difference  ^ 

4.  If  a  farmer  have  256  gallons  of  cider, 
how  many  barrels  holding  36  gallons  can  he 
fill? 

5.  From  a  cistern  holding  743  gallons,  98^ 
gallons  were  pumped  out  and  afterwards  39 
gallons  poured  in  ;  how  many  gallons  were 
then  in  the  cistern  ? 

6.  What  cost  37  carriages  at  $185  each  .^ 

7.  If  a  ship  sail  7289  miles  in  37  days,. 
how  many  miles  does  she  sail  per  day  ? 

8.  A  miller  paid  $169  for  78  bushels  of 
wheat,  ^97  for  oats  and  $395  for  corn  ; 
what  did  he  pay  for  all  of  the  grain } 

9    From  the  sum  of  397  and  6798,  take 

69. 

See  Teachers'  Editon,  p.  152. 


FIRST  STEPS  AMONG  FIGURES.  IO5 

10.  The  difference  between  two  numbers 
is  347.  and  the  less  number  is  79,  what  is 
the  greater  number  } 

1 1.  A  man  died  leaving  ^5600,  of  which 
he  gave  his  wife  $2,8CK),  his  son  $900,  one 
daughter  $850  and  the  rest  to  another 
daughter  ;  how  much  did  the  second  daugh- 
ter receive  .'* 

12.  A  man  bought  75  sheep  at  one  time, 
and  169  at  another  ;  he  sold  86  of  them  to 
one  man  and  49  to  another  ;  how  many 
had  he  left  > 

13.  Mr.  Wilson  bought  one  house  for 
^4150,  and  afterward  another  for  $3750  ;  he 
sold  both  of  them  for  $7000  ;  did  he  gain 
or  lose,  and  how  much  .'* 

14.  There  is  an  orchard  consisting  of  24 
rows  of  trees,  and  36  trees  in  each  row  ; 
how  many  apples  in  the  orchard,  allowing 
an  average  of  2079  on  a  tree  ? 

15.  A  man  owing  $7165,  gives  in  pay- 
ment 39  cows  valued  at  ^48  each  and  $750 
in  money  ;  how  much  does  he  still     ^-^  ' 

16.  Add  16  thousand  20,  fifty  millio*    '»«^ 
thousand  nine,  79  thousand  847,  and  9  n*. 
lion  79  thousand  8. 

17.  How  many  tons  of  hay  at  $18  a  ton 
must  be  given  for  16  horses  at  $153  each? 

18.  639+91. 758+9.347  +  81.731 -i-9.34^ 
+  35,446+8.237+12,849+87,677=  ? 


104  FIRST  STEPS  AMONG  FIGURES. 

19.  A  grocer  spent  $881  for  molasses 
and  sugar ;  he  paid  $368  of  the  money  for 
molasses,  and  the  rest  for  27  barrels  ot 
sugar;  how  much  did  the  sugar  cost  a 
barrel  ? 

20.  How  mai/y  yards  of  cloth  in  6S  bales, 
•each  bale  having  97  pieces,  and  each  piece 
containing  29  yards  ? 

2 1  Paid  $6  each  for  75  sheep,  and  sold 
the  flock  for  $400  ;  did  I  gain  or  lose,and 
how  much  ? 

22.  How  many  horses  at  $165  each  can 
*>e  bought  for  $2360? 

23.  How  much  is  gained  by  buying  48 
-cows  at  $37  each,  and  selling  them  at  $45 
each  ? 

24.  Mr.  Dixon  has  225  acres  of  land 
worth  $97  an  acre,  and  Mr.  Taft  has  196 
acres  worth  $79  an  acre  ;  how  many  acres 
have  the  two  together,  and  what  is  the 
value  of  the  whole  ? 

25.  A  man  sold  a  farm  of  96  acres  at  $g 
3r  if  -  :,  and  with  the  money  received  for  it 
^'"  ^nt  a  farm  of  135  acres  ;  what  did  he 
^4y  an  acre  tor  the  latter  farm  ? 

26.  A  teacher  had  his  life  insured  for 
/2500.  At  the  time  of  his  death  he  owned 
a  house  and  lot  worth  $1850  and  furniture 
ft^orth  $475.    He  owed  debts  to  the  amount 


FIRST  STEPS  AMONG  FIGURES.  I05 

of    ^^369 ;    how   much    did    he   leave   his 
family  ? 

27.  There  are  5280  feet  in  a  mile ;  how 
many  feet  in  709  miles  ? 

28.  A  man  starts  from  New  York  on 
Tuesday  morning  and  travels  at  the  rate  of 
57  miles  a  day  ;  another  starts  from  the 
same  place  Wednesday  morning  and  follows 
on  at  the  rate  of  69  miles  a  day  ;  how  far 
apart  are  they  Thursday  night .'' 

29.  James  sold  a  grocer  96  eggs  at  1$ 
cents  a  dozen,  and  received  120  cents  ;  how 
much  does  the  grocer  still  owe  him  ? 

30.  If  there  were  365  days  in  each  year, 
how  many  years  would  there  be  in  31390 
days  ? 

31.  Add  seventy  million  nine  hundred 
thousand,  two  hundred  six  thousand  eight, 
sixty  thousand  sixty,  seven  thousand  nine 
hundred,  ten  million  ten  thousand  ten,  and 
seven  hundred  fifty-nine  million  two  hun 
dred  thirty  thousand. 


I06  FIRST  STEPS  AMONG  FIGURES. 


I.  7657     2 

.  3456 

3  7567 

4775 

5767 

4756 

7777 

7475 

3675 

3456 

4567 

6777 

6735 

7756 

7SA^ 

5677 

5647 

5735 

7756 

6775 

6767 

3457 

7564 

7476 

6574 

4677 

3456 

5767 

3456 

3456 

4575 

5767 

7763 

7717 

7534 

4567 

5676 

n't  aT. 

6467 

2475 

2345 

5 

4-  4756 

•  7654 

7577 

4775 

6645 

5467 

7734 

7556 

5675 

6775 

6756 

3457 

7467 

5747 

4575 

7674 

5647 

6577 

7777 

7754 

6452 

7777 

3567 

7777 

6776 

4564 

5643 

5675 

See  Teachers'  Edition,  p.  153. 


FIRST  STEPS  AMONG   FIGURES.  I07 


6.  456S 

7.  6758 

8.  8765 

3785 

8547 

4658 

8678 

7868 

7777 

5786 

8778 

3456 

8678 

8888 

6778 

8888 

8888 

7l<^7 

8888 

3576 

8878 

5678 

6758 

77^7 

8765 

8687 

8585 

4876 

5678 

7777 

5487 

6786 

3456 

8888 

10.  3748 

8888 

8675 

7654 

7887 

4765 

4567 

7777 

8888 

7777 

8888 

5678 

4567 

8765 

7777 

4567 

7777 

8778 

4565 

7658 

8486 

Io8  FIRST  STEPS  AMONG  FIGURES. 


II.  6758 

12.  7658 

13.  7865 

7584 

4576 

4578 

4676 

6785 

8657 

^77^ 

5467 

678s 

3456 

8878 

8888 

8888 

4657 

8888 

8888 

8765 

3456 

6547 

6578 

7777 

3754 

7857 

5678 

6678 

5686 

•    8753 

7563 

8578 

4576 

5885 

67ZS 

8687 

4. 4786 

15.  7684 

8657 

4578 

6578 

3456 

7865 

S^Z^ 

4576 

^^^^ 

8888 

7654 

3456 

3567 

7777 

7777 

8765 

5648 

3578 

7385 

8657 

4637 

4768 

8386 

See  Teachers' 

Edition,  p.  154. 

FIRST  STEPS  AMONG  FIGURES,  IO9 


16.  3687  17 

.  4837 

18.  4587 

8546 

7584 

8635 

7685 

8758 

6754 

4868 

5875 

8888 

5784 

8888 

8888 

7777 

8888 

6754 

6548 

6753 

7578 

3675 

4584 

4785 

8888 

7777 

7777 

8888 

5678 

5768 

5678 

8765 

8654 

8563 

3857 

4585 

5785 

6586 

7848 

8678 

8465 

5686 

19.  8476 

20, 

.  7586 

5768 

4767 

4567 

8658 

7777 

4875 

7777 

8888 

8654 

8888 

•  3568 

7654 

8888 

3867 

8888 

7777 

4567 

4685 

8765 

8568 

4478 

7777 

7586 

* 

4825 

88^7 

8674 

no  FIRST  STEPS  AMONG  FIGURES. 

1.  4,570.365-323»456=? 

2.  9»374.056-636,587=? 

3.  685,700,365-296.314,537=? 

4.  76.400,235-3,234.567=? 

5.  38,500.684-8,769,876=? 

6.  7.460,683-379,876=? 

7.  375»6oo,735-83.735,829=? 

8.  83,640,574-5,712,653-=? 

9.  794.600.435-63.732,367=? 

10.  74.300.375-53.620.547=? 

11.  8.750.043-970,236=? 

12.  6,713,021-6,873.213=? 

13.  48,300.563-9,000,687=? 

14.  487.500,564-65.730,637=? 

15.  756,000,375-85.203.456=? 

16.  6,847,000,346-367,020,654=? 

17.  79,068x58  =  ? 

18.  80,479x74=? 

19.  4,185x368=? 

20.  968,579x798=? 

21.  79.689x4,759=? 

22.  79,867x6.897=? 

23.  85,765  X  8 1.072=? 

24.  49.678x9,876=? 

25.  497^896x8,659=? 

See  Teachers'  Edifion.  p.  154- 


FIRST  STEPS  AMONG    FIGURES.  HI 

Since  moving  a  figure  one  place  to  the 
left  increases  its  value  ten  fold,  and  moving 
it  two  places,  ten  times  ten  fold  or  one 
hundred  fold, — to  multiply  any  number  by 
10,  lOO,  looo,  &c.,  annex  as  many  ciphers 
to  the  multiplicand  as  there  are  in  the 
ihultiplier. 

379  X  icx)=37,900  Ana. 

26.  7,865  X  10,000=? 

27.  573  X  io=? 

28.  68  X  ioo=? 

29.  6,320  X  1000=? 

30.  875  X  100,000=? 

When  there  are  ciphers  at  the  right  of 
either  the  multiplier  or  multiplicand,  or  of 
both,  place  the  multiplier  under  the  multi- 
plicand so  that  the  significant  figures 
farthest  to  the  right  shall  come  under  each 
other.  After  multiplying  by  the  significant 
figures  and  adding,  write  as  many  ciphers 
at  the  right  of  the  product  as  there  are  at 
the  right  of  the  multiplier  and  multiplicand 
together. 

(These  directions  are  given  very  minutely 
but  are  not  to  be  committed  to  memory.) 


112  FIRST  STEPS  AMONG  FIGURES. 


34200 

34000 

1368 

1026 

1,162,800,000 

31.  3,750x46,000=? 

32.  46,300x350=? 

33-  635,000x700-? 

34.  27,500x680,000=? 

35.  586,000x7,400=? 

36.  490x36,700=? 

37.  6,840x7.500=? 

38   8,609x800=? 

39.  67,900x870=? 

40.  8.690x4.700=? 

41.  480.600x7,090=? 

42.  70,580x6,408,000=? 

43.  706,900x5,078,000=? 

44.  68,090  X  70,900=? 

45.640,980x10,000=? 

I.  43,188,278-^9=? 

2.    791.071,117^9=? 


FIRST  STEPS  AMONG  FIGURES.  II3 


3.  6,156,712,635-4-8  =  ? 

4.  67,815,232^7  =  ? 

5.  6,928,288,028-9  =  ? 

6.  52.717.437^8=? 

7.  8,808,273,807^9=? 

8.  461,392,186-4-6=? 

9.  49.347765-7=? 
10.  44,160,343^-9=? 

Ji-  557793.576^7=? 

12.  7,664,063,843^8=? 

13.  8,044,i85,6o7-=-9=? 

14.  461,093,406^7=? 

15.  5,225,741^6=? 

16.  529^762,735-7=? 

17.  7,230,245^8-? 

18.  86,274,817-^9=? 

19.  716.863,843-4-8=? 

20.  41,088.317^7=? 

21.  30,884,751-^-7.058=? 

22.  34,600.073^-6,032  =  ? 

23.  1,890,186-^5,178  =  ? 

24.  1,203,161,896^8,169=? 

25.  5.279.490-814  =  ? 

26.  33,620,328-4-725=? 

Sec  Teachers'  Edition,  p.  155. 


114  FIRST  STEPS  AMONG  FIGURES. 

27.  92.197.364^6,257  =  ? 

28.  7,638,482^439  =  ? 

29  511,764.908-7,461=? 

30.  946,526,656-6,397  =  ? 

31.  65.790.555-^6,847  =  ? 

32.  0,363,666^469  =  ? 

33.  18,827.247-4-378  =  ? 

34.  526.493.286^-3,794  =  ? 

Since  moving  a  figure  one  place  to  the 
right  diminishes  its  value  ten  fold,  and 
two  places,  ten  times  ten  fold  or :  one 
hundred  fold,— to  divide  any  number  by 
10,  100,  1000,  &c.,  cut  off  by  a  vertical 
line  as  many  figures  on  the  right  of*  the 
dividend  as  there  are  ciphers  at  the  right 
of  the  divisor. 

The  nun^ber  at  the  left  of  the  vertical 
line  will  be  the  quotient,  and  the  number  at 
the  right  of  it  the  remainder. 

Illustration  :  78634-^100=? 

Solution  :  786  |  34  the  quotient  is  786 
and  34  is  the  remainder. 

1.  793,468     10,000  =  ? 

2.  37,680^100  =  .'* 

3.  2,347,600^100,000=? 
4    76,219,648-5-100=.^ 

5.  372,938,641-^10,000,000=? 


FIRST  STEPS  AMONG  FIGURES.  11$; 

To  divide  by  any  number  with  ciphers  at 
the  right. 

78»673^700=? 

Divide  both  dividend  and  divisor  by  lOO, 
and  cutting  off  the  2  figures  at  the  right,  and 
the  example  becomes — 

7|oo)786i73 

112—273  rem.,  or 

112 — 

700 

Divide  the  number  at  the  left  of  the 
vertical  line  in  the  dividend,  by  the  number 
at  the  left  of  the  vertical  line  in  the  divisor, 
and  to  the  remainder  annex  the  figures  of 
the  dividend  cut  off. 


45|ooo)6i2|37o(i3  quo. 
45 

162 

135 

27370  rem. 


i;i6  FIRST  STEPS  AMONG  FIGURES. 

Hence  the  rule  :  To  divide  by  any  num- 
ber with  ciphers  at  the  right,  cut  off  the 
ciphers  ^t  the  right-  of  the  divisor  by  a 
vertical  line,  and  also  as  many  figures  at  the 
right  of  the  dividend.  Divide  the  remain- 
ing number  in  the  dividend  by  the  remain- 
ing number  in  the  divisor,  and  to  the 
remainder  annex  the  figures  cutoff  from  the 
rightof  the  dividend  for  the  true  remainder. 

1.  18,228,211 -5-37,500=.!* 

2.  5,142,762,131-750,000=? 

3.  546,927,300-^687,000=? 

4.  70,514,152^796,800=? 

5.  8,734.758^10,000=? 

6.  350.870,000^3,580,000=? 

7.  3,278,300-^7,000=? 
S.  87. joo  X  23,000=? 
9.  7,162,323-900  =  ? 

10.  394.690.750-5,800=? 

11.  29,850,010^3.750=? 

12.  From  six  billion  six  thousand  six, 
take  eighty  million  eight. 

13.  27,752,320,172-^570,000=? 

14.  2,910,144,700-7-36,800=? 

See  Teachers'  Edition,  p.  155. 


FIRST  STEPS  AMONG  FIGURES.  I  I  7 

15.  178,576,495-- 100,000  =  ? 

16.  365,820,038-6,000  =  ? 

17.  475,308,056-48,600  =  ? 

18.  Subtract  forty-five  million,  one 
thousand  ten,  from  forty-two  million  seven 
hundred  thousand. 

19.  441.937.000-^597.000=- 

20.  30,500,857,231^3,780,000=? 

21.  3,657,200^600=? 

22.  47,096  X  8,600=? 

23.  70,286,631^900=? 
24  7960  X  100= ? 

When  the  multiplier  is  less  than  13  the 
pupil  should  be  taught  and  required  to 
multiply  but  once  through,  multiplying  by 
II  or  12  as  he  has  already  been  taught  to 
multiply  by  4,  5  or  6. 

1.  78,967  X  i2=? 

2.  69,789/  II  =? 

3.  754.836  X  I2  =  ? 

4.  845,768  X  I2  =  ? 

5.  68,094,796  X  1 1  =? 

6.  586,974  X  I2  =  ? 

See  Teachers'  Edition,  p.  157. 


1 1 8  FIRST  STEPS  AMONG  FIGURES. 

7.  6,579.687  X  I2  =  ? 

8.  69,487,968  X  I2  =  ? 

9.  9,468.579  X  II=? 

10.  479,658  X  I2  =  ? 

11.  97.5^9.647^  I2  =  ? 

12.  6.975,897  X  12  =  ? 

13.  69.786995  X  I2  =  ? 

14.  94,679.689x12=? 
IS-   979.896  X  I2  =  ? 

16.  685.796  X  II=? 

17.  8.798,979  X  I2  =  ? 

18.  49,897,697  X  I2  =  ? 

19.  97,987,986  X  I2  =  ? 

20.  7,989,985  X  12  =  ? 

21.  1,185,491,377-7-12  =  ? 
'22.    546,569.461-5-11=? 

23.  820,499,872-^-12  =  ? 

24.  1,044,922,315-^12  =  ? 

25.  9.438,575.969-12  =  ? 

26.  9,568.867,509-^11=? 

27.  455.976,730^12  =  ? 

28.  956,219,625-7-12  =  ? 

29.  898,436,885-^12=? 

30.  7.434.306,992-^11=? 


FIRST  STEPS  AMONG  FIGURES.  II9 

FOR    ORAL    RECITATION. 

1.  A  bov  having  25  cents,  bought  mar- 
bles at  4  cents  each,  keeping  5  cents  of  the 
money  ;  how  many  marbles  did  he  buy  ? 

2.  Jane  lost  10  cents  on  her  way  to  the 
post-office,  and  spent  the  rest  of  her  money 
for  10  3-cent  stamps:  how  much  money 
had  she  when  she  started  ? 

3.  *  Mark  earned  8  cents,  lost  5  cents,, 
and  then  found  10  cents,  when  he  had  25 
cents  ;  how  much  money  had  he  at  first  ? 

4.  What  cost  12  pounds  of  sugar  if  y 
pounds  cost  63  cents  ? 

5.  A  man  spent  $s^  ^^^en  earned  $7, 
and  after  giving  away  ;^6  found  he  had  $15  ; 
how  many  dollars  had  he  at  first  ? 

6.  If  6  apples  cost  2  cents,  what  cost  i& 
apples  ? 

Solution:  If  6  apples  cost  2  cents,  iS 
apples,  which  are  3  times  6  apples,  will  cost 
3  times  2  cents  or  6  cents. 

7.  If  4  marbles  cost  3  cents,  what  cost 
24  marbles .? 

•  If  he  had  25  cents  a/t^r  hnding  10  cents,  before  he 
found  it  he  had  the  difference  between  25  cents  and  10 
cents,  or  15  cents.  If  he  had  15  cents  a//^r  losing  5 
cents,  ^/art  he  lost  it  he  had  the  sum  of  15  cents  and  5 
cents  or  20  cents.  If  he  had  20  cents  a/ttr  earning  » 
cents,  before  he  earned  it  he  had  the  difference  betweea 
ao  cents  .ind  8  cents  or  12  cents. 

Sec  Teachers'  Edition,  p.  161. 


120  FIRST  STEPS  AMONG  FIGURES. 

8.  What  cost  20  figs,  if  5  figs  cost  2 
cents  ? 

9    If  3  oranges  cost  12  cents,  what  cost 
7  oranges  ? 

10.  What  cost  12  lemons,  if  5  cost  25 
cents  ? 

1 1  How  many  apples  can  be  bought  for 
15  cents,  at  the  rate  of  5  for  3  cents  ? 

12.  If  3  oranges  cost  10  cents,  how  many 
may  be  bought  for  40  cents } 

13.  30  cents  will  buy  how  many  apples, 
at  9  for  6  cents  ? 

14.  How  many  figs  may  be  bought  for 
24  cents,  at  the  rate  of  3  figs  for  2  cents? 

15.  If  4  marbles  cost  5  cents,  what  cost 
20  marbles  ? 

16.  If  5  lemons  cost  20  cents,  what  cost 
9  lemons  ? 

1 7.  What  cost  30  pears  if  3  pears  cost  5 
cents? 

18.  If  4  peaches  cost  3  cents,  what  will 
24  peaches  cost  ? 

19  At  the  rate  of  2  oranges  for  9  cents, 
how  many  may  be  bought  for  18  cents? 

20.  If  3  men  cut  6  cords  of  wood  in  a 
•day,  how  many  cords  will  7  men  cut  in  a 
day  ? 

21.  If  3  men  dig  a  ditch  in  12  days,  how 
long  will  it  take  one  man  ? 

See  Teachers'  Edition,  p.  162. 


FIRST  STEPS  AMONG  FIGURES.  121 

22.  If  4  men  harvest  a  field  of  wheat  in 

8  days,  how  many  days  will  it  take  i   man 
to  harvest  it  ? 

23.  If  12  men  can  dig  a  field  of  potatoes 
in  13  days,  how  many  men  will  do  it  in  i 
day? 

24.  How  many  men  can  load  a  car  in  i 
hour,  if  2  men  can  load  it  in  4  hours  ? 

25.  If  4  men  can  do  a  piece  of  work  in 
12  days,  how  long  will  it  take  3  men  to  do 
it? 

26.  How  many  days  will  it  take  6  men 
to  earn  $32,  if  it  takes  4  men  6  days  to 
earn  it  ? 

27.  If  4  men  can  do  a  piece  of  work  in 

9  days,  how  many  men  can  do  it  in  6  days  ? 

28.  If  6  men  can  do  a  piece  of  work  in 
4  days,  how  many  men  will  it  take  to  do 
the  work  in  3  days  ? 

29.  If  6  men  can  do  a  piece  of  work  in 
12  days,  how  long  will  it  take  4  men  to  do  it  ? 

30.  If  4  men  can  build  12  rods  of  wall 
in  a  day,  how  many  rods  can  6  men  build 
in  a  day  ? 

31.  How  many  men  will  build  a  wall  in 
12  days,  if  6  men  build  it  in  8  days  ? 

32.  If  3  men  cut  7  cords  of  wood  in  a 
day,  how  many  cords  will  12  men  cut  in  a 
day? 


122  FIRST  STKPS  AMONG  FIGURES. 

33.  A  girl  took  7  pins  from  a  paper  and 
then  put  on  9 ;  her  brother  afterwards  took 
off  6,  and  left  in  it  24 ;  how  many  on  the 
paper  at  first  ? 

34  How  many  men  will  do  a  work  in  6 
days  that  9  men  do  in  4  days  ? 

35.  If  8  men  do  a  work  in  6  days,  how 
many  men  will  do  it  in  12  days  ? 

36.  If  4  men  do  a  work  in  12  days,  bow 
long  will  it  take  6  men  ? 

37.  If  3  pencils  are  worth  1 1  cents,  how 
many  pencils  can  be  bought  for  33  cents  ? 

38.  A  girl  having  a  paper  of  candy,  ate 
7  pieces  ;  then  her  brother  gave  her  5  pieces, 
after  which  she  gave  her  mother  9  pieces. 
She  had  left  27  pieces ;  how  many  pieces 
had  she  at  first  ? 

39.  What  number  divided  by  2  will  give  6 } 

40.  If  4  cords  of  wood  cost  $20,  how 
many  cords  can  be  bought  for  $35  ? 

41.  If  6  vests  are  worth  $24,  what  are 
9  vests  worth  ? 

42.  If  5  cords  of  wood  cost  ^24,  what 
will  1 5  cords  cost  ? 

43.  What  number  divided  by  3  will  get 
12? 

44.  At  10  cents  a  pint,  what  will  a  gallon 
of  molasses  cost  ? 

45.  How  many  bushels  of  potatoes  at  4 


FIRST  STEPS  AMONG  FIGURES.  1 23 

shillings  a  bushel  may  be  bought  for  3 
bushels  of  wheat  at  12  shillings  a  bushel? 
46  A  boy  gave  8  marbles  worth  2  cents 
apiece,  for  7  pencils  worth  3  cents  each  ; 
how  much  did  he  gain  ? 

47.  How  many  eight-gallon  cans  will  be 
required  to  hold  56  gallons  of  milk  ? 

48.  If  15  bushels  of  wheat  will  make  3 
barrels  of  flour,  how  many  bushels  will 
make  8  barrels  ? 

49.  How  many  yards  of  cloth  at  $6  a 
yard  will  pay  for  9  tons  of  coal  at  $S  a  ton  ? 

50.  When  flour  is  $7  a  barrel,  how  many 
barrels  can  be  bought  for  $8,  and  9  bushels 
of  wheat  at  $3  a  bushel  ? 

51.  96  eggs  are  how  many  dozen  ? 

52.  If  8  horses  eat  48  bushels  of  oats  in 
.2  weeks,  how  many  bushels  will  5  horses 

eat  in  the  same  time  ? 

53.  Our  school  has  a  recess  in  the  fore- 
noon and  also  in  the  afternoon.  If  there 
are  one  hour  of  school  before  each  recess 
and  two  hours  after  each  recess,  how  many 
hours  of  school  in  a  week  ? 

54.  If  two  apples  cost  one-half  of  10 
cents,  how  many  can  be  bought  for  15  cents  ? 

55.  How  many  three-cent  stamps  can  be 
bought  for  27  cents  ? 

56.  A  boy  caught  some  fishes  ;  he  threw 


124  FIRST  STEPS  AMONG  FIGURES. 

away  7,  then  caught  3  and  bought  2,  when 
he  had  14;  how  many  did  he  catch  at 
first? 

57.  Kate  lives  two  miles  from  school, 
and  does  not  go  home  at  noon ;  how  far 
must  she  walk  in  a  week  if  she  loses  no 
time  at  school  ? 

58.  Frank  has  6  five-cent  pieces,  4  three- 
cent  pieces  and  five  two-cent  pieces  ;  how 
many  cents  has  he  ? 

59.  John  has  7  cents,  his  brother  8,  and 
their  sister  has  4  more  than  both  of  them ; 
how  many  have  they  all  ? 

60.  Which  costs  the  more,  3  lemons  at 
4  cents  each  or  6  pears  at  2  cents  each  ? 

61.  A  boy  went  to  the  grocery  with  25 
cents,  and  bought  2  pounds  of  sugar  at  9 
cents  a  pound  ;  how  much  change  should  he 
bring  back  if  he  has  2  cents  for  doing  the 
errand  ? 

62.  If  2  barrels  of  flour  will  last  3  men 
6  months,  how  long  will  it  last  9  men  ? 

63.  Bought  some  peaches  for  24  cents, 
at  the  rate  of  5  for  2  cents,  and  divided 
them  equally  among  6  boys ;  how  many 
did  each  boy  receive  ? 

64.  If  7-  bushels  of  clover  seed  are  worth 
I42,  how  many  bushels  of  wheat  at  $2  a 
bushel  will  3  bushels  of  clover  seed  buy  ? 


FIRST  STEPS  AMONG  FIGURES. 


65.  If  2  men  start  from  the  same  place 
and  travel  in  the  same  direction,  one  6 
miles  an  hour,  and  the  other  3  miles  an 
hour,  how  far  apart  will  they  be  in  9  hours  ? 

66.  In  how  many  hours  will  a  man  who 
drives  8  miles  an  hour  overtake  a  lootman 
who  is  60  miles  ahead,  and  walks  at  the 
rate  of  3  miles  an  hour  ? 

67.  *A  man  bought  a  span  of  horses  for 
$100,  paid  S6o  for  their  keeping,  ajid  sold 
them  for  $200 ;  what  did  he  gain  on  each 
horse  ? 

68.  How  many  turkeys  can  I  buy  for 
$43,  at  the  rate  of  3  for  $5,  and  have  $S  left  ? 


EXAMPLES  FOR  THE  SLATE. 

1.  In  a  certain  church  28  pews  rent  at 
$35  each,  19  at  $23  each  and  37  at  $1$ 
each  ;  for  how  much  do  they  all  rent  ? 

2.  A  railroad  18  miles  long  cost  ^452, 
682  for  labor,  and  $177,228  for  other  ex- 
penses ;  what  was  the  cost  per  mile  ? 

3.  One  half  of  the  inhabitants  of  Con- 
stantinople are  Turks,  150,000  Greeks, 
30,000  Armenians,  and  65,000  Jews;  how 
many  in  all  ? 

See  Teachers'  Fdition.  p.  165. 


126  FIRST  STEPS  AMONG  FIGURES. 

4.  If  a  man  earns  $960  a  year,  and 
spends  yearly  $688,  in  how  many  years 
will  he  lay  up  $4,624. 

5.  A  man  bought  a  farm  for  $17,600  ; 
he  sold  half  of  it  for  $9,322,  at  the  rate  of 
$79  an  acre ;  how  many  acres  did  he  buy  ? 
How  much  did  he  pay  an  acre  ? 

6.  From  the  sum  of  7574  and  10746, 
take  their  difference. 

7..  A  lady  having  $125,  paid  $37  for  a 
set  of  furs,  and  $2  a  yard  for  23  yards  of 
silk  ;  how  much  money  had  she  left  ? 

8  13341*6  emigrants  arrived  in  New 
York  in  1867,  which  was  9.731  more  than 
arrived  in  1866  ;  how  many  arrived  in  1866? 

9  James  and  George  started  together, 
and  traveled  in  the  same  direction.  James 
walked  2  miles  an  hour  and  George  4  miles 
an  hour  ;  how  far  apart  were  they  at  the  end 
of  19  hours  ? 

10  In  six  boxes  of  crayons  there  are  864 
pieces  ;  if  864  pieces  cost  360  cents,  what 
will  one  box  cost  ? 

11.  There  are  two  numbers,  the  greater 
of  which  is  37  x  96,  and  their  difference  is 
18  X  27  ;  what  are  the  numbers  ? 

1 2.  A  earns  $45  a  month,  and  B  earns  13 
times  as  much  lacking  $490 ;  how  much  does 
B  earn  in  8  months  ? 


FIRST  STEPS  AMONG  FIGURES.  1 27 

13.  If  a  house  is  worth  $i,8oo  and  the 
farm  on  which  it  stands  five  times  as  much 
lackinj^  $36,  and  the  stock  one-third  as 
much  as  the  house  and  farm,  what  is  the 
value  of  the  whole  ? 

14.  A  man  sold  his  farm  of  245  acres  at 
$69  an  acre  and  bought  some  land  at  $97 
an  acre  ;   how  many  acres  could  he  buy  ? 

15.  Mr.  Smith  was  968  miles  from  home; 
he  traveled  toward  home  137  miles  one  day; 
119  the  next  day,  and  98  the  third  day; 
how  far  was  he  from  home  then  ? 

16.  From  thirty  billion  ten  thousand, 
take  seven  billion  two  hundred  nine  thou- 
sand seventy-five. 

17.  A  man  bought  325  bushels  of  barley 
for  $500  ;  450  bushels  of  oats  for  $250  ; 
625  bushels  of  corn  for  $150  more  than  he 
paid  for  the  oats  ;  300  bushels  of  beans  at 
$2  a  bushel,  and  some  wheat  for  ;^ioo  more 
than  he  paid  for  the  corn  ;  how  much  did 
he  pay  for  all  ? 

18.  How  many  solid  feet  of  earth  can  be 
removed  in  36  days  by  two  carts  each 
carrying  9  loads  a  day,  and  34  solid  feet  at 
a  load  > 

19.  A  man  having  $9,840,  gave  each  of 
his  two  sons  $2,750  and  the  remainder  to 
his  daughter ;  how  much  did  he  give  his 
daughter  ? 


128  FIRST  STEPS  AMONG  FIGURES. 

20.  $26,250  is  3  times  what  A  gave  for 
his  farm,  and  he  gave  $370  more  for  it  than 
it  was  worth;  how  much  was  the  farm 
worth  ? 

21.  I  sold  a  horse  for  $375  which  cost  me 
$29$  ;  how  much  did  I  gain  ? 

22.  I  sold  a  cow  for  $65  and  by  so  doing 
lost  $15;  what  did  she  cost  ? 

23.  A  man  began  business  with  $3,850, 
and  in  7  years  he  was  worth  $10,465  ;  how 
much  did  he  make  each  year  ? 

24.  How  many  days  would  36  horses  live 
on  an  amount  of  food  that  would  keep  24 
horses  288  days  ? 

25.  A  merchant  received  $248  on  Mon- 
day and  $396  on  Tuesday ;  what  was  the 
average  receipts  per  day  ? 

26.  Two  men  start  from  the  same  place 
and  travel  in  opposite  directions,  one  at  the 
rate  of  54  miles  a  day,  and  the  other  at  the 
rate  of  45  miles  a  day  ;  how  far  apart  wilJ 
they  be  at  the  end  of  6  days?  How  far 
apart  if  they  travel  in  the  same  direction  ? 

27.  A  man  bought  478  bushels  of  corn  ; 
all  but  1 36  bushels  were  sunk  in  a  boat ; 
how  much  was  saved  ? 

28.  A  merchant  bought  46  yards  of  cloth 
for  $93,  and  sold  it  at  $3  a  yard ;  how  much 
did  he  gain  ? 

See  Teachers'  Edition,  p.  166. 


FIRST  STEP.^  ...»i«7.>^.  FIGURE^S.  I  29 

29.  Divide  the  product  of  79  and  237 
by  their  difference. 

30.  At  $135  each,  how  many  horses 
can  be  bought  for  $9,368  ? 

31.  How  many  times  can  317  be  sub 
tracted  from  13,314? 

32.  There  are  3  bins  containing  856 
bushels  of  wheat  ;  i  contains  376  bushels, 
another  contains  297  bushels  ;  how  many 
in  the  third  bin  ? 

33.  A  farmer  sold  1 3  tons  of  hay  at  $  16  a 
ton,  and  24  cords  of  wood  at  $$  2l  cord  ;  he 
divided  the  money  received  among  four  credi- 
tors ;  how  much   money  did  each  receive  ? 

34  A  has  18  barrels  of  flour  of  196 
pounds  each  ;  it  a  family  of  9  persons  use 
49  pounds  of  flour  a  week,  how  long  will 
the  flour  last  them  ? 

35.  If  Mr.  Long's  sheep  were  put  into 
6  fields,  96  in  a  field,  there  would  be  5 
sheep  remaining  ;  how  many  sheep  has  he  ? 

36  A  grocer  bought  2  cheeses,  one 
■weighing  68  pounds  and  the  other  75 
pounds,  at  14  cents  a  pound;  how  many 
cents  would  he  gain  by  selling  them  at  17 
cents  a  pound  .' 

37.  A  man  killed  four  hogs,  one  weigh- 
ing 368  pounds,  one  412,  one  379  and  one 
433  ;  what  was  their  average  weight  ? 


130  FIRST  STEPS  AMONG  Hv.uKhS. 

38.  There  were  84  sheep  in  fourptisturesp 
.  there  were  30  in  the   first  and  24  in  the 

second;  if  there  were  an  equal  number  irk 
each  of  the  others,  how  many  in  each  ? 

39.  If  a  man  paid  $500  for  four  horses, 
$200  for  5  cows  and  $175  for  40  sheep^ 
how  many  animals  did  he  buy  ? 

40.  If  a  man  earns  $685  a  year,  and 
spends  $496  a  year,  in  how  many  years 
will  he  save  $1,134? 

41.  How  many  pounds  of  coffee  at  2J 
cents  a  pound  will  pay  for  three  hogsheads- 
of  sugar,  each  containing  1080  pounds,  at 
12  cents  a  pound  ? 

42.  What  is  the  sum  of  the  difference 
and  sum  of  1768  and  987  ? 

43  A  man  deposited  in  bank  at  different 
times  $397,  $450  and  $568  ;  he  drew  out 
at  one  time  $275  and  at  another  $368  ; 
how  much  remained  in  the  bank  ? 

44.  A  man  sold  26  cows  at  $35  each; 
how  many  horses  at  $145  each  can  he  buy 
with  the  money  received  ? 

45.  A  dealer  shipped  500  bushels  of 
beans  in  250  bags,  600  bushels  of  wheat  ir> 
280  bags ;  he  used  136  less  bags  in  which 
to  ship  3(X>  bushels  of  corn  than  he  did  for 
the  wheat:  he  put  1200  bushels  of  oats  in 
bags  holding  2  bushels  each  ;  how  many 
bags  did  he  use  for  all  the  grain  ? 


FIRST  STEPS  AMONG  FIGURES.  13/ 

46.  The  income  of  a  man  who  *  struck 
oil "  is  $75  a  day ;  how  many  teachers 
would  this  employ  at  $850  a  year  ? 

47.  A  farmer  having  $1397,  bought  9 
tons  of  hay  at  $16  a  ton,  a  horse  for  $185, 
155  sheep  at  $4  each,  and  spent  the  rest  of 
his  money  for  cows  at  $32  each  ;  how  many 
cows  did  he  buy  ? 

48.  A  fisherman  caught  2  dozen  fishes  ; 
he  sold  one-half  of  them  at  25  cents  each  ; 
the  other  half  for  26  cents  each,  except  one 
which,  weighing  33  pounds,  he  retailed  at  1 1 
cents  per  pound ;  how  much  did  he  get  for 
his  fishes  ? 

49.  A  boy  paid  100  cents  for  5  quires  of 
paper  (24  sheets  each)  and  sold  it  at  the 
rate  of  2  sheets  for  3  cents  ;  did  he  gain  or 
lose,  and  how  much  ? 

50.  How  many  half  dimes  in  350  cents? 
51    A  miller  ground  34  bushels  of  wheat, 

18  of  corn,  and  22  of  oats  ;  how  many  bags 
holding  2  bushels  each,  held  the  grain  ?  What 
did  the  grinding  cost  at  7  cents  a  bushel  ? 
52.  24  boys  attended  the  same  school, 
but  in  three  different  rooms  ;  5  were  in  one 
room,  and  8  in  another,  and  if  the  number 
of  boys  in  the  third  room  be  multiplied  by 
12,  the  product  will  equal  the  number  of 
blackbirds  they  saw  on  their  way  to  school ; 
bow  many  did  they  see  ? 


132  FIRST  STEPS  AMONG    FIGURES. 

53  Sound  travels  at  the  rate  of  1090 
feet  in  a  second ;  at  this  rate  how  long 
would  it  take  the  report  of  a  cannon  to 
reach  the  moon,  which  is  240.000  miles 
away  (i  mile  is  5280  feet)  ? 

54.  An  estate  of  $14350  was  divided 
between  a  widow  and  two  children  ;  the 
widow'?;  share  was  55450,  the  son's  $1280 
less  than  the  widow's,  and  the  daughter 
had  the  rest ;  how  much  did  the  daughter 
have  ? 

55.  The  product  of  two  numbers  is 
36288.  and  one  of  them  is  756 ;  what  is  the 
other  ? 

56  A  man  bought  145  acres  of  land  for 
^,850,  and  95  more  acres  at  $45  an  acre  ;  he 
sold  the  whole  at  $$6  an  acre  ;  did  he  gain  or 
lose,  and  how  much  ? 

57.  A  farmer  bought  47  acres  of  land 
for  54.416,  and  34  acres  at  $75  an  acre  ; 
what  was  the  average  price  per  acre  ? 

58.  The  sum  of  two  numbers  is  7568. 
and  one  of  them  is  784  ;  what  is  the  other? 

59  How  many  military  companies  of  98 
men  each,  can  be  formed  from  7,463  men  ? 

60.  How  many  yards  of  cloth  at  24 
cents  a  yard,  will  pay  for  26  dozen  eggs  at 
14  cents  a  dozen,  and  ajar  of  butter  worth 
2^4  cents  ? 


FIRST  STEPS  AMONG  FIGURES.  I33 

61.  George  and  Lewis  start  from  the 
same  place  at  the  same  time,  and  travel  in 
the  same  direction,  George  at  the  rate  of 
714  rods  an  hour,  and  Lewis  at  the  rate  of 
579  rods  an  hour;  how  far  apart  are  they 
at  the  end  of  9  hours  ?  How  far  apart  in 
7  hours,  if  they  had  traveled  in  opposite 
directions  ? 

62.  If  I  receive  $40  a  month  and  spend 
$32  a  month,  in  how  many  years  will  I 
save  $1,152  ? 

63.  Subtract  the  difference  between  79 
and  2300  from  their  sum. 

64.  What  is  the  sum  of  ten  thousand 
ninety,  seven  thousand  nine  hundred,  eight 
million  nine  hundred  eighteen,  five  hun- 
dred thousand,  seventy  thousand  seventy- 
five,  and  eight  hundred. 

65  The  dividend  is  736592,  the  divisor  is 
6978  ;  what  is  the  quotient  and  remainder.? 

66.  The  remainder  is  658  and  the  sub- 
trahend 1734  ;  what  is  the  minuend  ? 

Pupils  make  and  solve  the  following  ex- 
amples : 

67.  Given  a  multiplicand  of  4  figures,  a 
multiplier  of  3  figures,  required  the  product  ? 

68.  Given  the  minuend  and  remainder, 
find  the  subtrahend. 

69.  Given  the  subtrahend  and  the  re- 
mainder, find  the  minuend. 


134  FIRST  STEPS  AMONG  FIGURES. 

70.  Given  the  sum  of  three  numbers  and 
two  of  them,  find  the  third.  ^ 

71.  Given  the  difference  between  two 
numbers  and  the  less  number,  find  the 
greater. 

72.  Given  the  divisor,  quotient  and  re- 
mainder, find  the  dividend. 

73.  Given  the  product  of  two  numbers 
and  one  of  them,  find  the  other. 

74.  Given  the  difierence  between  two 
numbers  and  the  greater  number,  find  the 
less  number. 

75.  Given  whole  price,  number  of  arti- 
cles, find  the  price  of  a  different  number 
of  articles. 

76.  Given  the  cost  and  selling  price,  find 
the  gain. 

TJ.  Given  the  selling  price  and  the  loss, 
find  the  cost 

7S.  Given  the  cost  and  the  gain,  find 
selling  price. 

79.  Given  the  selling  price  and  gain,  find 
cost. 

80  Given  the  cost  and  the  loss,  find  the 
selling  price. 


FIRST  STEPS  AMONG  FIGURES.  I  35 


I.  6745 

2.  5847 

3-  9337 

5678 

9576 

4598 

9867 

4684 

3765 

6543 

5967 

9458 

7698 

8439 

7^95 

4759 

4785 

4739 

5978 

9478 

5345 

9647 

4567 

4869 

8458 

8975 

3765 

4796 

4687 

479« 

4.  8451 

5.  5747 

6759 

9835 

7846 

4696 

9567 

8739 

6976 

5684 

7569 

7938 

8427 

6456 

9568 

8845 

3753 

7587 

5679 

8946 

36  FIRST  STEPS  AMONG  FIGURES. 


d  5768 

7.  4768 

8.  5869 

9576 

9535 

9768 

4869 

9849 

4352 

8753 

7697 

4675 

4968 

4857 

9538 

7495 

9564 

4678 

^478 

5739 

5765 

4956 

5647 

4976 

9875 

9788 

9738 

5637 

5347 

4657 

8592 

8234 

9876 

^  7648 

10.  7465 

3752 

5876 

9746 

4795 

3859 

8649 

8432 

3578 

9594 

9756 

3745 

6549 

9458 

8732 

4637 

5685 

8358 

7839 

3543 

5693 

FIRST  STEPS  AMONG  FIGURES.  137 


II.  4538   i: 

2.  9638 

13.  9768 

9764 

5796 

759^ 

3797 

4579 

4879 

8436 

8947 

9687, 

9768 

6896 

4739 

4593 

9758 

8645 

8945 

4563 

3787 

7657 

9755 

9568 

4895 

6879 

4837 

7537 

5768 

8795 

4885 

8597 

4568 

5938 

4856 

975<^ 

9457 

9589 

6897 

14.  6795 

15.  5896 

4569 

9748 

8769 

7635 

9345 

5864 

8876 

9787 

6789 

4538 

3954 

7687 

8456 

4859 

3789 

• 

5321 

8375 

4978 

9999 

9654 

3478 

7987 

6457 

9868 

138  FIRST  STEPS  AMONG  FIGURES. 


L  6587 

17.  9768 

18.  4879 

4759 

4593 

9763 

5896 

7846 

4598 

9537 

5937 

8756 

4678 

9876 

4975 

4957 

2345 

6874 

8436 

3527 

7589 

6758 

8498 

4837 

4957 

6549 

5692 

8796 

7856 

7859 

4587 

7387 

4537 

7948 

9765 

6895 

6595 

3849 

4576 

4769 

9674 

3844 

9487 

3758 

7989 

5896 

6847 

4596 

FIRST  STEPS  AMONG  FIGURES.  139 


19-  4532 

20.  9476 

7856 

3869 

4978 

5458 

9456 

7567 

3279 

4835 

4856 

8769 

7995 

7654 

8447 

8579 

9568 

6432 

4789 

4976 

6435 

9845 

7896 

5637 

4967 

7948 

8538 

8654 

7689 

6739 

9876 

479« 

Dnvls  (W.  W.)  FRACTIONAL  APPARATUS,  consisting  of 
eight  wooden  balls,  three  inches  in  diameter,  one  whole, 
and  the  others  divided  respectively  into  halves,  thirds, 
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PP  43 

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Fit  ell    (Joshu.i  G.)  TJis  AH  of  Questioning.     Second   Edition. 

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Heiidriek  (Mary  F.)  A  8erie*  of  OuesOonsin  English  and  Ameri- 
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